5. From a pool of 16 candidates, 9 men and 7 women, the offices of presiden vice president, secretary, and treasurer will be filled. (a) In how mar different ways can the offices be filled? (b) What is the probability that four of the offices are filled by women?

To tackle your question, let’s break it down into two parts! First, to find the total ways to fill the four offices (president, vice president, secretary, treasurer) from the pool of 16 candidates, you can think of it as a permutation since the roles are distinct. The number of ways to select and arrange 4 out of 16 candidates is calculated as \( P(16, 4) = 16 \times 15 \times 14 \times 13 = 43680 \). Now, for the probability that all four offices are filled by women, we first identify the number of ways to choose 4 women from the 7 available. This would be calculated as \( P(7, 4) = 7 \times 6 \times 5 \times 4 = 840 \). The probability is the number of favorable outcomes (all women in office) divided by the total outcomes. Hence, the probability is \( \frac{840}{43680} = \frac{1}{52} \). Now you've got both parts: the total arrangements and the intriguing probability! Enjoy solving more puzzles!