How many ways can you pick a president vice-presidential and secretary from a group of 6 boys and 5 guys if a) there must be at least 1 boy chosen and b) there must be only 1 girl chosen?

Answer 1

To solve this, we first determine the number of ways to select a president, vice president, and secretary from a total of 11 individuals (6 boys and 5 girls).

For part (a), ensuring at least 1 boy is chosen can be framed as the total arrangements minus the arrangements with no boys. The arrangements where no boys are chosen (only girls) are not possible since we need at least one boy.

For part (b), choosing exactly 1 girl means selecting 1 from 5 girls and then selecting the remaining 2 positions from the 6 boys. The arrangement can be done in ( \binom{5}{1} \times (6^2) \times 3! ) ways, where ( 3! ) accounts for the arrangement of the selected individuals.

Calculating this gives ( 5 \times 36 \times 6 = 1080 ) ways for part (b). Thus, the answer is ( 1080 ) ways with 1 girl chosen and 2 boys chosen.