Given:
Number of girls = 20
Number of boys = 30
The president must be a girl and the vice president a boy
To find:
Number of ways to choose a president, vice president, and secretary.
Solution:
The president must be a girl and the vice president a boy. So, out of three students 1 is girl and 1 is boy. Third student can be a girl or a boy.
Total number of ways = Selecting 2 boys and 1 girl + Selecting 1 boy and 2 girls
[tex]=^{30}C_2\times ^{20}C_1+^{30}C_1\times ^{20}C_2[/tex]
[tex]=\dfrac{30!}{2!(30-2)!}\times \dfrac{20!}{1!(20-1)!}+\dfrac{30!}{1!(30-1)!}\times \dfrac{20!}{2!(20-2)!}[/tex]
[tex]=\dfrac{30\times 29\times 28!}{2\times 1\times 28!}\times \dfrac{20\times 19!}{19!}+\dfrac{30\times 29!}{29!}\times \dfrac{20\times 19\times 18!}{2\times 1\times 18!}[/tex]
[tex]=\dfrac{30\times 29}{2}\times 20+20\times \dfrac{20\times 19}{2}[/tex]
[tex]=8700+3800[/tex]
[tex]=12500[/tex]
Therefore, the required number of ways is 12500.
