Combinations (the number of subsets)

A set in mathematics is a group/collection of distinct things/elements/members.
No duplicates.  No order (it's like a bag).
{a,b,c,d,e} a set of 5 elements

A subset of a set is any collection of any of the elements of the set.
{a,d,e}  {b}   {c,a,e,d}  are some subsets of the above set.
committees, subgroups, samples

How many subsets of a particular size k are there of a set of size n?
nCk = n! / (n-k)!k!

5C3 = 5! / (5-3)!3! =                                 120 / 12 = 10
               by hand  = 5*4*3*2*1 / 2*1*3*2*1 = 5*2

{a,b,c}
subsets of size 1 (singletons): {a} {b} {c}    3C1 = 3
subsets of size 2 (pairs): {a,b} {a,c} {b,c}   3C2 = 3

{a,b,c,d}
subsets of size 1 (singletons): {a} {b} {c} {d}                 4C1 = 4
subsets of size 2 (pairs): {a,b} {a,c} {a,d} {b,c} {b,d} {c,d}  4C2 = 6
subsets of size 3 (triples): {a,b,c} {a,b,d} {a,c,d} {b,c,d}    4C3 = 4

Either by hand, calculator, or 
https://davidwills.us/math103/combos_perms.html

5C1 = ________
5C2 = ________
5C3 = ________
5C4 = ________

10C1 = ________
10C2 = ________
10C3 = ________
10C4 = ________
10C5 = ________
10C6 = ________
10C7 = ________
10C8 = ________
10C9 = ________

52C5 = _____________   the number of 5-card hands from a deck of 52 cards.

52C13 = _______________  the number of 13-card hands from a deck of 52 cards.
                   Each one you get is probably the first time it's ever been dealt.

100C10 = ___________________
In a group of 100 people, there are this many subsets of 10 people. A lot.
A data sample is a subset of the population. Even for a small population
there is a huge number of possible samples. For large populations, the
number of samples of n=30 or n=100 is practically infinite.


{x,y,z} subsets of size 2:___________________________
{w,x,y,z} subsets of size 2:___________________________
{w,x,y,z} subsets of size 3:___________________________
{v,w,x,y,z} subsets of size 2:_______________________________________________

{v,w,x,y,z} subsets of size 3:_______________________________________________

{v,w,x,y,z} subsets of size 4:_______________________________________________


Pascal's triangle
each row: first and last is 1. "interiors" are sum of the two diagonally above

row#                                                      row sum  == 2^row
----                                                      -------     -----
 0                               ___
                              ___  ___
                            ___  ___  ___
                         ___  ___  ___  ___
                       ___  ___  ___  ___  ___
                    ___  ___  ___  ___  ___  ___
                  ___  ___  ___  ___  ___  ___  ___
               ___  ___  ___  ___  ___  ___  ___  ___
...

Row n is the nCk's

nCk = n-1Ck-1 + n-1Ck   (that's how you filled in the triangle)
6C3 = 5C2 + 5C3
___ = ___ + ___

8C6 = __C__ + __C__ = ___ + ___ = ______
8C4 = __C__ + __C__ = ___ + ___ = ______