How many ways a committee consisting of 5 men and 6 women can be formed from 8 men and 10 women?

Discussion :: Permutation and Combination - General Questions (Q.No.8)

Satya said: (Aug 18, 2010)
I'm not getting the second step of solution. How it became (8C5 x 10C6) to (8C3 x 10C4)?

Arpita said: (Aug 31, 2010)
Can someone tell how the second step conversion happen?.

Farjana said: (Sep 28, 2010)
Actually the formula for combination is nCr = nC(n-r). Therefore 8C5 = 8C(8-5) = 8C3.

Sundar said: (Sep 28, 2010)
Hi Farjana, Thanks for your valuable information. Have a nice day!

Decom said: (Mar 7, 2011)
Does this assume that the men/women can be in any order or have to be allocated a position in the group. I felt this was a little ambiguous.

Saloni said: (May 1, 2011)
Why there is a * sign instead it should be +?

Nagarajan said: (May 30, 2011)
May anyone exlplain why that conversion is required, nCr = nC(n-r). but I know it is the formula.. Please explain conceptually.

Sathya said: (Jun 24, 2011)
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 !!!!

Deepu2543651 said: (Aug 24, 2011)
Why we getting wrong answer while we directly apply nCr formula?

Musa said: (Sep 15, 2011)
I don't understand the formula used. Can somebody explain it?

Rohini said: (Sep 15, 2011)
Why we have to convert nCr = nC (n-r) ?

Sunita said: (Sep 17, 2011)
Its for simplicity you can do it by solving first step also.

Priyadarshini said: (Jan 28, 2012)

nCr=nC(n-r) is used for simplification purpose. We can also calculate it directly as nCr.

Answer will be same in both ways.

Joe said: (Feb 6, 2012)
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...

Senorita said: (Feb 19, 2012)
As per my knowledge d calculation above shows only selection part. Aren't we suppose to calculate the number of ways it can b arranged into? in this case 11! ?

Diwakar said: (Apr 25, 2012)
Formula: n!/r!(n-r)! 8C5 = 8!/5!(8-5) which is 678/123 = 56 10C6 = 10!/6!(10-6)! which is 78910/1234 = 210 Therefore: 56*210 = 11760

Manu said: (Mar 30, 2013)
Yes, If you do it with (nCr = nC (n-r) ) then also we are getting the same answer. Please don't get confuse.

Rockstr said: (Jul 18, 2013)
Why are we not multiplying with 11! ? these 5 men and 6 women can also be arranged within themselves. Right?

Shuhaib said: (Oct 9, 2013)
Another method. In unit place we can fix only one number ie 5. Other place we can selected 6c2. So answer is 6c2*1.

Vimala said: (Nov 5, 2013)
No need to multiply with 11! because they did not ask for the arrangement of the person. We need to concentrate only on the selection of person for the committee.

Younisali.Syed said: (Oct 16, 2014)
I agree with this answer but when you see question no: 4 in that he selected and then arranged in this problem he only selected. Any one tell me why so?

Flair said: (Jan 6, 2015)
In this question don't give importance to arrangements. Like ABC=ACB=BAC they are same. Take this example with person.

Siva said: (Jan 24, 2015)
Different arrangement of digits in a number gives different numbers, but arrangement doesn't alter the composition of a committee. So no need to calculate no.of arrangements. All arrangements give the same committee.

Nishat Parwez said: (May 19, 2015)
The correct formula for the mentioned question is n!/r!( n-r)!.. This makes the concept clear. Totally agree with @Diwakar.

Vishesh Bakshi said: (Jun 5, 2015)
Am always confused whether to multiply or add in between this two cases. How to determine it correctly can any one help me out on this?

Poonam said: (Jun 25, 2015)
Do not use the formula which does not exist for actual solution. That formula is for permutation which you given. Please explain in another way.

Bala said: (Jul 12, 2015)
Ya its confusing please tell in simple way.

Priya said: (Sep 30, 2015)
When I'm solving it directly like 8C5 and 10C6 means I'm getting wrong answer.

Karthika said: (Oct 15, 2015)
Why we are applying the ncr=n-r formula in the second step? What is the purpose of using where we should we use this?

Webuzzi said: (Nov 24, 2015)
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.

Ravi said: (Feb 27, 2016)
Why they reduce it to 8C3 and 10C4?

Ganganesh said: (Jul 2, 2016)
Thank you all.

Tshering said: (Aug 5, 2016)
@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.

Raghav said: (Sep 6, 2016)
8C5 * 10C6 = 11760. But an addition to this 11760 * 11!? Why is this missing?

Akash said: (Nov 9, 2016)
Apply permutation formula when the order in which you place the object doesn't matter. Permutation formula is : nPr = n! / (n-r)! * r! . This is what is simply done here.

Lester said: (Nov 29, 2016)

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.

Shiv said: (Dec 19, 2016)

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.

Rishu.R said: (Mar 18, 2017)
Why the second step conversion is needed? Is it necessary? How do I predict the 2nd step conversion from the question? Anyone, please help me. In detail.

Toluwalase said: (Mar 25, 2017)
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.

Siri said: (Apr 19, 2017)
@Toluwalase. This doesn't talk about the arrangement. It talks about the selection.

Charan said: (Aug 23, 2017)
8c3 = 8!/3!(8-3)! = 8!/3!5! which is equal to 8c5 = 8!/5!(8-5)! = 8!/5!3!.

Charan said: (Aug 23, 2017)
6men and 5 women can also be changed so it can be 11!*11760.

Gayudhaya said: (Sep 25, 2017)
In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there? By using above formula just explain it.

Tejal said: (Jan 6, 2018)
Don't we have to divide (8C5*10C6) by total no of selection that is 18C11?

Nancy said: (Mar 3, 2018)
Please help me to understand why the deduction (the second step) is done?

Pankaj Kumar said: (May 15, 2018)
The second step involves the formula of nCr, which is; nCr = n!/(n-r)!r!.

B S said: (May 24, 2018)
The Answer should be 266.

Shivani Bairagi said: (Aug 17, 2018)
Why we take 8C3 * 10C4? I am not understanding. So please explain me in detail.

Rahul said: (Sep 2, 2018)
I am not getting this answer. Please, anyone, help me to get this.

Mohan said: (Jul 24, 2019)
8C5 = (87654)/(54321). = (876)/(321). = 8C3. 10C6 = (1098765)/(654321). = (10987)/(4321), = 10C4.

Sayali said: (Jul 30, 2019)
nCr=n!/r! (n-r)! =8C5=8! /5!* 3! = 56. & 10C6=10! /6! 4! =210. Therefore, 56210 = 11760.

Prakash said: (Oct 17, 2019)
Kindly anyone tells the difference between permutations and combinations.

Rahul said: (Nov 28, 2019)
But, this is the formula of permutation, so why we convert?

Shubhadhayinee said: (Dec 13, 2019)
What @Farjana said is absolutely right.

Lalit said: (Feb 18, 2020)
As per @Farjana said, but we didn't apply this formula for previous examples see example -1.

Siddhant B said: (Jun 17, 2020)
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.

Tsherin said: (Aug 28, 2020)
@Sayali and @Diwakar. We know the formula for Combination: nCr= n!/(n-r)!r! as mentioned. But why we aren't using r? Please explain.

Maheen said: (Jun 10, 2021)
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.