Aptitude - Permutation and Combination - Discussion

@Reena.
Why Internal Arrangements Are Not Considered:

Nature of a Committee:
A committee is typically a group of individuals where order doesn't matter.

For example:
If we select A, B, C, D, E as the 5 men, this is the same as selecting B, A, C, D, E, or any other arrangement of these names. The focus is only on who is in the group, not the sequence.

Focus on Selection, Not Order:
The task is to determine how many ways we can select 5 men and 6 women. Since their roles or hierarchy aren't specified, the internal order of people in the committee is irrelevant.

Combination Formula Used:
​ This ensures we don’t double-count the same group in different orders.

Key Point: The committee is just a collection of members, so the order in which they are chosen or arranged doesn't matter.

I think, yes it is true that 8C5 is equal to 8C3, as well as 10C6 to 10C3, because the operation in the denominator is multiplication, in 8C5, the denominator r! (n-r) ! is 5! (3!) , while in 8C3, the denominator is 3! (5!) , either way you will get the same answer, but in relation to the problem above, you are choosing 5 men out of 8 men, that is 8C5, and it is wrong to interpret it as 8C3, because that notation represents choosing 3 men out of 8 men. Though the answers are the same, the two notations have a different meaning.

I think that provides an explanation.

Yeah, I think I get the second step of the solution. But why do we have to find the number of ways the others can be arranged. We were asked to find the number of ways 5 men could be gotten from a total of 8 men and how we can get 6 women from the total of 10 women to form the committee. So the answer is meant to be 266. Why the additional calculation? Please answer.

It can also be calculated by simply calculating the combination of 8C5 men and 10C6 women. The answer would be 56 and 210, respectively.

And lastly, multiplying the combination of both men and women, which would be 11760. The second step which is n-r is quite confusing to some.

Simply use the formulae of combination and you'll get your answer.

By nCr.

8C5 * 10C6 = (8 * 7 * 6 * 5 * 4/1 * 2 * 3 * 4 * 5)* (10 * 9 * 8 * 7 * 6 * 5 * 4/1 * 2 * 3 * 4 * 5 * 6),
= 56 * 210,
= 11760.

Similarly, nC (n-r).

8C3 * 10C5 = (8* 7 * 6/1 * 2 * 3) * (10 * 9 * 8 * 7/1 * 2 * 3 * 4),
= 56 * 210,
= 11760.

Why all should try this formula you will still get the right answer n!/(n!-r!)r!.

8!/(8!-5!)5! which will yield 56 then use same formula and solve for 10 combination 6 it will yield 210. Then multiply the former result by the later result. That is 56 multiplied by 210 will yield 11760.

Actually the formula for combination is nCr = nC(n-r).

Therefore 8C5 = 8C(8-5) = 8C3.
the answers will come the same , simply it will make the problem easier
If the combination e.g is 11C4 then = 11C(11-4) = 11C7 it will also come the same answers with 11C4...

Why we didn't use this (nCr = nC(n-r)) formula in the following questions ?

http://www.indiabix.com/aptitude/permutation-and-combination/discussion-688
http://www.indiabix.com/aptitude/permutation-and-combination/discussion-684

Please explain me !!!!

The question says men and women, not men or women.

So when we see "and" we use * not +. If the question was men or women then the answer would have been 56+210. But in this case, its men and women so its multiplication i.e 56*210 = 11760.

@Ravi.

8C5 = (8 * 7 * 6* 5 * 4)/(5 * 4 * 3 * 2 * 1) = after cancelling 5 & 4, left with (8 * 7 *6 ) / (3 * 2* 1 ) which is same as 8C3.

8C3 = (8 * 7* 6 )/(3 * 2 * 1).

Another way of 8C5 coming to 8C3 is 8C5 = 8C(8-5) = 8C3.