Find the number of combinations of the for objects ABCD take

Find the number of combinations of the for objects A,B,C,D taken three at a time . How many committees of three can be formed from eight people.

Solution

We\'ll write the formula of the combination of n elements taken k at a time:

C(n,r) = n!/k!(n-k)!

We\'ll establish that each combination consists of 3 objects.

We\'ll have 3! permutations of objects in the combination.

We\'ll note the permutation as P.

P = 3!

P = 1*2*3

P = 6

The number of combinations will be multiplied by 3!:

C(4,3) = P(4,3)/3!

P(4,3) = 4*3*2

P(4,3) = 24

C(4,3) = 24/6

C(4,3) = 4

The possible combinations are:

C(4,3) = {abc , abd , acd , bcd}

To determine the number of committees of three that can be formed from eight people, we\'ll apply the combination formula:

C(8,3) = 8!/3!(8-3)!

C(8,3) = 8!/3!*5!

But 8! = 5!*6*7*8

3! = 1*2*3

C(8,3) = 5!*6*7*8/1*2*3*5!

We\'ll simplify and we\'ll get:

C(8,3) = 7*8/1

C(8,3) = 56 committees of three that can be formed from eight people.