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There are 3 Female Players, 2 Male Players, 6 Female Singers, 5 Male Singers, 2 Female Actors and 4 Male Actors. For a Celebrity show you have to create a judge panel of 4 members (1 Head Judge & 3 Supporting Judges). Head judge should be a Player.No Player can be a support judge. If There should be at least 1 Female Singer and 1 Male Singer as supporting judges. What are the total number of ways for choosing them?

My approach

1 Head Judge- 5C1

1 Female Singer - 6C1

1 male Singer - 5C1

any other - (among 5FS 4MS 2FA 4MA) - 15C1

Such that for the last position there could either be Singers or Actors.

So I'm getting $5*6*5*15=2250.$

Given Solution

1 Head Judge / 1 Female Singer / 1 Male Singer / Any Actor $= 5C1 .6C1.5C1.6C1 =900$

1 Head Judge / 2 Female Singer / 1 Male Singer $= 5C1.6C2.5C1 =375$

1 Head Judge / 1 Female Singer / 2 male Singer $= 5C1.6C1.5C2 =300$

Total ways$= 1575$

What is wrong with my argument? Thanks in advance.

emil
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    What is wrong with your solution is that you are doublecounting scenarios where a singer picked in step2 or step3 is instead picked during step4. Consider a smaller example where you have only two people total... both of them are female singers... and we want to have two judges such that at least one of them is a female singer. Of course the only way to do this would be to have them both be the judges and this can be done in only one way, but by your logic you have $2$ ways to pick the necessary singer, and then you pick the remaining singer later during the "any other" step. – JMoravitz May 11 '18 at 07:40
  • They said that at least 1 female and male singer must be present ... But you are not counting when they are more than one... – Nehal Samee May 11 '18 at 07:53
  • @NehalSamee, he did count that – Rhys Hughes May 11 '18 at 08:11

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