Download A combinatorial formula:

A combinatorial formula:
How many groups (combinations) of 4 letters can you make from the 5 letters
ABCDE?
ABCD
ABCE
ABDE|
ACDE
BCDE
Answer: 5 different groups or combinations
How many groups (combinations) of two letters can you make?
AB
AC
AD
AE
BC
BD
BE
CD
CE
DE
Answer:10 different groups or combinations
How many groups (combinations) of 1 letter ?
A
B
C
D
E
Answer 5 different groups or combinations
In general:
given n objects, the number of combinations of r objects is denoted by C(n,r) and
computed as:
C(n,r) = n!/r!(n-r)!
(alternate notation)
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For example,
C(5,4) = 5!/4!1! = (5*4*3*2*1)/(4*3*2*1) * 1 = 4
C(5,2) = 5!/2!3! = (5*4*3*2*1)/(2*1 )*( 3*2*1) = 10
C(5,0) = 5!/0!5! = 1 -- Note 0! = 1 by definition
A shortcut:
Find C(6,2):
Notice that after cancellation:
Here is another example. Notice the pattern.
How many different 5 card poker hands are there:
One way: C(52,5) = 52!/5!47! = 2588960
After all the cancellation:
C(52,5) = 52*51*50*49*48/ 5!
Also,
C(50,3) = 50*49*48/3! (after cancellation)
C(n,r) = n*n-1…n-r+1/r! ( The numerator has r factors)
C(20,6) = 20*19*18*17*16*15 / 6! ( numerator has 6 factors)
C(5,0) = 1
C(4,2) = 4*3/2*1 = 6
How many ways can you fill out a lottery ticket where you choose 6 numbers
from 36?
C(36,6) = 36*35*34*33*32*31/6! = 36*35*34*33*32*31/6*5*4*3*2*1 = 1,947,792
So you chance of winning is 1/1947792 = .000000534
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