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Combinatorics¶
Combinatorics is a branch of mathematics dedicated to figuring out how to count things! Specifically, the number of elements contained in – i.e., the cardinality of – event \(A\):
Counting’s not so hard, you say? Think again: when it comes to ordering and choosing sets in complicated and clever ways counting can get tricky in heartbeat. Be that as it may, combinatorics plays a fundamental and foundational role in probability as it forms the basis for assigning probabilities to events in many contexts. Beyond that, the need to count carefully and correctly is obviously a hugely important part of Data Science, and it’s important to have few of the “standard counting tricks” handy or you might make something a whole lot harder than it needs to be. For a good “industry perspective” on the importance and challenge of counting in data science check out Counting in Data Science.
Factorials¶
Factorials count the number of ways to order a set of objects.
E.g., if there are 10 lottery balls (labeled 1-10) and we draw them all, how many possible orderings could be drawn? The answer to this question is
- there are 10 choices for the first ball
- 9 choices for the second ball (because we’ve already drawn the first lottery ball)
- 8 choices for the third ball (because we’ve already drawn the first two lottery balls)
- and so on...
until there is only one ball left and we must pick it. That is, there are \(10*9*8*\cdots*1 = 10!\), i.e. 10 factorial, possible orderings. This is a demonstration of the so-called _product rule_ for counting things, and it forshadows the incredibly fundamental rule in probability known as the _chain rule_ (which we will return to later).
The number 10 factorial can be calculated in Python, but watch out: factorials get really big really fast...
>>> import math
>>> math.factorial(10)
3628800
Combinations¶
Combinations count the number of ways to choose things when order does not matter. Here’s an example of all the two character combinations that can be made from the letters A, B, and C:
>>> from itertools import combinations
>>> list(combinations("ABC",2))
[('A', 'B'), ('A', 'C'), ('B', 'C')]
The “number of combinations” problem – i.e., counting the number of all possible unordered collections of size K from a pool of N objects — is often referred to as the “N choose K” problem, and the solution to the problem is commonly notated as
Note
EXERCISE
If you think about the “N choose K” formula carefully you can actually see that it makes sense: the \(K!\) in the denominator is the number of ways to order a list of length \(K\), and the \(\frac{N!}{(N-K)!}\) is all possible lists of length \(K\) where order matters. Do you see why? Knowing this, can you articulate why the complete formula counts the right thing?
If you’re feeling like this seems like an awful lot of multiplication and division, don’t worry, Python can do all the necessary calculations for you:
>>> from math import factorial
>>> def comb(n, k):
... return factorial(n) / (factorial(k) * factorial(n - k))
>>> from scipy.misc import comb
>>> comb(3,2)
3.0
Note
EXERCISE
>>> lefthand_beers = ["Milk Stout", "Good Juju", "Fade to Black", "Polestar Pilsner"]
>>> lefthand_beers += ["Black Jack Porter", "Wake Up Dead Imperial Stout","Warrior IPA"]
- We have sampler plates that hold 4 beers. How many different ways can we combine these beers?
- Print a list of these pairs so we can identify the bad ones?
Source: lefthandbrewing.com/beers
Permutations¶
Permutations counts the number of ways subsets can be chosen when order does matter. If you followed the “N choose K” thought exercise above then you won’t be surprised to learn (i.e., be reminded) that the number of ways to choose K things out of N things when order matters is
Explicitly writing out the formula makes it clear that permutations are just a slight variation on what we did for the product rule above. I.e., they are just a special case of factorial multiplication. And of course, once again, it’s easy to Python take care of the permutation calculations for you.
from math import factorial
def permu(n, k):
return factorial(n) / factorial(n - k)
>>> from itertools import permutations
>>> list(permutations("ABC",2))
[('A', 'B'), ('A', 'C'), ('B', 'A'), ('B', 'C'), ('C', 'A'), ('C', 'B')]
Note
EXERCISE
On a baseball team with 12 players, how many different batting lineups are there?
Hint: there are 9 players in a lineup.
Note
PAIRED EXERCISE
Face off against the person next to you, pitting permutations against combinations, and defend the dignity and honor of your cause! Which is your favorite and why, when it really comes down to it? No low blows allowed, but don’t be afraid to bring size into the argument if you think it helps your case.


Gather slices from params axis axis according to indices.
tf.compat.v2.gather
tf.gather( params, indices, validate_indices=None, axis=None, batch_dims=0, name=None )
Gather slices from params axis axis
according to indices
. indices
must
be an integer tensor of any dimension (usually 0-D or 1-D).
For 0-D (scalar) indices
:
output
$$[p_0, ..., p_{axis-1}, \hspace{5.1em} p_{axis + 1}, ..., p_{N-1}]$$ =\params
$$[p_0, ..., p_{axis-1}, \hspace{1em} indices, \hspace{1em} p_{axis + 1}, ..., p_{N-1}]$$.
For 1-D (vector) indices
with batch_dims=0
:
output
$$[p_0, ..., p_{axis-1}, \hspace{2.6em} i, \hspace{2.6em} p_{axis + 1}, ..., p_{N-1}]$$ =\params
$$[p_0, ..., p_{axis-1}, \hspace{1em} indices[i], \hspace{1em} p_{axis + 1}, ..., p_{N-1}]$$.
In the general case, produces an output tensor where:
$$\begin{align} output[p_0, &..., p_{axis-1}, & &i_{B}, ..., i_{M-1}, & p_{axis + 1}, &..., p_{N-1}] = \ params[p_0, &..., p_{axis-1}, & indices[p_0, ..., p_{B-1}, &i_{B}, ..., i_{M-1}], & p_{axis + 1}, &..., p_{N-1}] \end{align}$$
Where $$N$$=ndims(params)
, $$M$$=ndims(indices)
, and $$B$$=batch_dims
.
Note that params.shape[:batch_dims] must be identical to
indices.shape[:batch_dims].
The shape of the output tensor is:
output.shape = params.shape[:axis] + indices.shape[batch_dims:] + params.shape[axis + 1:]
.
Note that on CPU, if an out of bound index is found, an error is returned. On GPU, if an out of bound index is found, a 0 is stored in the corresponding output value.
See also tf.gather_nd
.
params
: The Tensor
from which to gather values. Must be at least rank
axis + 1
.indices
: The index Tensor
. Must be one of the following types: int32
,
int64
. Must be in range [0, params.shape[axis])
.validate_indices
: Deprecated, does nothing.axis
: A Tensor
. Must be one of the following types: int32
, int64
. The
axis
in params
to gather indices
from. Must be greater than or equal
to batch_dims
. Defaults to the first non-batch dimension. Supports
negative indexes.batch_dims
: An integer
. The number of batch dimensions. Must be less
than rank(indices)
.name
: A name for the operation (optional).A Tensor
. Has the same type as params
.


|
JavaTM 2 Platform Standard Ed. 5.0 |
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public interface ORBInitInfoOperations
Passed to each ORBInitializer
, allowing it to
to register interceptors and perform other duties while the ORB is
initializing.
The ORBInitInfo
object is only valid during
ORB.init
. If a service keeps a reference to its
ORBInitInfo
object and tries to use it after
ORB.init
returns, the object no longer exists and an
OBJECT_NOT_EXIST
exception shall be thrown.
ORBInitializer
| Method Summary | |
|---|---|
void |
add_client_request_interceptor(ClientRequestInterceptor interceptor)
Used to add a client-side request Interceptor to the list of client-side request Interceptors. |
void |
add_ior_interceptor(IORInterceptor interceptor)
Used to add an IOR Interceptor to the list of IOR Interceptors. |
void |
add_server_request_interceptor(ServerRequestInterceptor interceptor)
Used to add a server-side request Interceptor to the list of server-side request Interceptors. |
int |
allocate_slot_id()
Called to allocate a slot on PortableInterceptor.Current . |
String[] |
arguments()
Returns the arguments passed to ORB.init . |
CodecFactory |
codec_factory()
Returns the IOP.CodecFactory . |
String |
orb_id()
Returns the ID of the ORB being initialized. |
void |
register_initial_reference(String id,
Object obj)
Identical to ORB.register_initial_reference . |
void |
register_policy_factory(int type,
PolicyFactory policy_factory)
Registers a PolicyFactory for the given
PolicyType . |
Object |
resolve_initial_references(String id)
Identical to ORB.resolve_initial_references . |
| Method Detail |
|---|
String[] arguments()
ORB.init
. They may or
may not contain the ORB's arguments.
String orb_id()
CodecFactory codec_factory()
IOP.CodecFactory
. The
CodecFactory
is normally obtained via a call to
ORB.resolve_initial_references( "CodecFactory" )
, but
since the ORB is not yet available and Interceptors, particularly when
processing service contexts, will require a Codec
, a
means of obtaining a Codec
is necessary during ORB
initialization.
void register_initial_reference(String id, Object obj) throws InvalidName
ORB.register_initial_reference
. This same
functionality exists here because the ORB, not yet fully initialized,
is not yet available but initial references may need to be
registered as part of Interceptor registration. The only difference
is that the version of this operation on the ORB uses PIDL
(CORBA.ORB.ObjectId
) and
(CORBA.ORB.InvalidName
) whereas the version in this
interface uses IDL defined in this interface; the semantics are
identical.
InvalidName
Object resolve_initial_references(String id) throws InvalidName
ORB.resolve_initial_references
. This same
functionality exists here because the ORB, not yet fully initialized,
is not yet available but initial references may be required from the
ORB as part of Interceptor registration. The only difference is that
the version of this operation on the ORB uses PIDL
(CORBA::ORB::ObjectId
and
CORBA::ORB::InvalidName
) whereas the version in this
interface uses IDL defined in this interface; the semantics
are identical.
This operation is only valid during post_init.
InvalidName
void add_client_request_interceptor(ClientRequestInterceptor interceptor) throws DuplicateName
interceptor
- The ClientRequestInterceptor
to be
added.
DuplicateName
- thrown if a client-side request Interceptor
has already been registered with this Interceptor's name.void add_server_request_interceptor(ServerRequestInterceptor interceptor) throws DuplicateName
interceptor
- The ServerRequestInterceptor to be added.
DuplicateName
- thrown if a server-side request Interceptor
has already been registered with this Interceptor's namevoid add_ior_interceptor(IORInterceptor interceptor) throws DuplicateName
interceptor
- The IORInterceptor to be added.
DuplicateName
- thrown if an IOR Interceptor has already
been registered with this Interceptor's name.int allocate_slot_id()
PortableInterceptor.Current
.
Note that while slot id's can be allocated within an ORB initializer, the slots themselves cannot be initialized.
BAD_INV_ORDER
- thrown, with a standard minor code of 14, if
set_slot
or get_slot
is called on the
PICurrent
within an ORB initializer.Current
void register_policy_factory(int type, PolicyFactory policy_factory)
PolicyFactory
for the given
PolicyType
.
type
- The policy type that the given PolicyFactory
serves.policy_factory
- The factory for the given policy type.
BAD_INV_ORDER
- thrown, with a standard minor code of 16, if
a PolicyFactory
already exists for the given
PolicyType
.
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Copyright © 2004, 2010 Oracle and/or its affiliates. All rights reserved. Use is subject to license terms. Also see the documentation redistribution policy.


Among some Ghanaian ethnic groups like the Akan, Ga, Ewe and Nzema, a name is automatically assigned based on the day the child is born. These day names correspond to the day of the week someone is born and so by default, everybody has one - though the name may not necessarily appear on official documents.
This is the naming culture of the people of Ghana
As we have seen from the text above and the naming culture of the Ghana people we know that you are wondering
what your name would be based on your gender and the day that you we born.
To know your name and your age you just need to fill in the following form and the system will automatically
tell your Akan name and the day of the week that you were born i.e Monday, Tuesday e.t.c
Make sure you provide the correct details for you to get your results.
You can reach us through all our social media platforms.
info@derraphics


Dragon Boat Museum / Li Xiaodong Atelier
- Curated by 韩爽 - HAN Shuang
- about 5 hours
Museums & Exhibit · Foshan, China
- version="1.0" encoding="UTF-8"? Architects: Li Xiaodong Atelier
- AreaArea of this architecture projectArea: 578.1 m2
- version="1.0" encoding="UTF-8"?YearCompletion year of this architecture project Year: 2018
- version="1.0" encoding="UTF-8"?PhotographsPhotographs: Wenji Liang, Mian Li, Qiucheng Zhu, Rong Fan, Yuyan Zhou
- Lead Architect Xiaodong Li
- Design Team Xiaogang Lian, Chenyang Zhang
More Specs Less Specs
Text description provided by the architects. The project site lies within the Shunfeng mountain park, embraced by picturesque lake and mountains.
To celebrate the tradition of dragon boat festival of the place, the architect has laid the primary volume of the museum above the lake, intending to create a poetic sense to bring human, architecture and nature together. Using steel as main structure for the two-storey building, the museum is in 50 meters long and 13 meters wide.
Though with strong and masculine language expressed by the steel structure resembling the energy of dragon boat race, the curvy veranda and façade has instilled elegance and serenity into the architecture and bring perfect harmony to the place.
The building façade is covered with steel cables connected to the planting troughs on top. This allows the vines to grow along the cables and eventually disguise the building in shades of green, intriguing passersby to discover the building. The concrete steps and ramps are linked with the existing circulation of the lake which bring visitors to the building and the deck.
The changing of material to durable WPC in low saturated colour marks the transition to the higher-level spaces. Furthermore, lighting features are hidden from sight, such as behind the staircase panel and grooves along the ramps, to optimize the natural lighting atmosphere.
The first floor of the museum is divided into a part that lies along the bank while another sitting on the lake. The platform along the bank is mainly finished with perforated steel panels that allows the plants to grow through, letting people feel the intimacy with the water body and the nature. Moving to the platform on the lake, both ends of the platform are connected with long timber planks right above the water to dock the dragon boats, diminishing the lines into the lake gracefully.
The spacious second floor of the building serves as the main functional space of the museum. In this room, there is a glass elevator that captures the scenic view of the surroundings. The space has timber strip ceiling and is connected to first floor and water platform with an external steel staircase as well. The middle part of the interior spaces is finished with Bolon carpet while the other parts that touch the wall are in walkable glass finish. Besides, modularized curtain walls are installed along the building elevations too. With this design, the visitors could enjoy the panoramic lake and mountain sceneries and the activities that happen below the building simultaneously.