Number of ways to make delegations

[songoku]

Homework Statement

Find number of ways to make delegations of 4 from 4 men and 5 women if there is at least one man and one woman

Relevant Equations

Permutation and Combination

I did it 2 ways:

(1) No. of ways = total ways - delegations of all men - delegations of all women = 9C4 - 4C4 - 5C4 = 120

(2) No. of ways = choosing one man x choosing one woman x choosing the other two people = 4C1 x 5C1 x 7C2 = 420

I am pretty sure method (1) is correct but I don't understand why method (2) is wrong

Thanks

 

[mjc123]

Method 2 counts some combinations more than once. E.g. if I first select man A and woman A, then from the remaining 7 take man B and woman B, the result is the same as if I first select man B and woman B, then from the remaining 7 take man A and woman A. (Or A/B then B/A, or B/A then A/B). But each of these counts separately in method 2.

 

[songoku]

Method 2 counts some combinations more than once. E.g. if I first select man A and woman A, then from the remaining 7 take man B and woman B, the result is the same as if I first select man B and woman B, then from the remaining 7 take man A and woman A. (Or A/B then B/A, or B/A then A/B). But each of these counts separately in method 2.

Thank you very much mjc123