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.

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neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

The number of combinations of  r things from n things is nCr = n!/((n-r)!r!

So the number of combinations of 3 things from 4 = 4C3 = 4!/((4-3)!3!

= 4*3*2*1)/(1*3*2*1) =1.

Therefore out of 4 persins A,B,C and D, the number  groups of 3 persons  is 3. 

Number of different committees of 3 from 8 people is 8C3 = 8!/(8-3)!3!= 8!/(5!3!) = 8*7*6/3*2*1 = 56 different committees.

Explanation

We get a group of 3 by leaving D and taking others

you get a diffrent group leaving C and taking others.

We get a different group by leavin B and taking others.

Similarly we get another different group by leaving A.

Thus 4 different ways.

Top Answer

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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.

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