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.