Question

Without using formulas, explain how you can determine each of the following:

c. If you have 10 people, how many different committees of 3 people can be formed for the offices of president, vice-president, and secretary?

d. Does order matter or not for part c? Explain.

Answer 1

Answer for C.):-

How many ways a commitee of 3 members can be formed from a group of 10 people?

Okay, so you have 10 peoples, and you are trying to find how many different committees of 3 you can make from this.

So for each committee, we have nine (10) choices for the first member. Once we have chosen the first member, we have eight (9) choices left for the second. Once we have chosen the first and second, we have seven (8) choices left for the third.

So the total combinations are 10 times 9 times 8. This gives us 720.

10*9*8=720

But this is not the answer! Because this would count each committee several times. If we have three students on a committee we can rearrange that committee several ways, but it is still the same committee. For a committee of three, we can put any of the three in the first slot, then any of the remaining two in the second slot, then the last remaining in the third slot. So each different committee can be arranged 3 times 2 times 1 different ways. 3*2*1=6. This gives us 6 different ways. So our total of 720 above counted each committee 6 different times!

So we need to divide our answer 720 by 6,

720/6= 120

and we find out we have a total of 120 actual different committees

10!/(7!3!)

= 10 * 9 * 8 / (3 * 2 * 1)

= 720 / 6

= 120

Answer for d):-

Order doesn't matter for this formation of committee as this the problem of combinations. And we have studied that for combinations order doesn't matter

By definition of combination: When the order doesn't matter, it is a Combination

And we have also seen in the problem, we from the committee of 3 people from 10 peoples without considering the orders of 10 people..

Final answer is for combinations Order doesn't matter.