Question

three research departments have 10, 7, and 6 members, respectively. each department is to select a delegate and an alternate to represent the department at a conference. in how many ways can this be done?

Explanation:

Step1: Calculate ways for first department

The number of ways to select a delegate and an alternate from a department of 10 members is the number of permutations of 10 taken 2 at a time. Using the permutation formula $P(n,r)=\frac{n!}{(n - r)!}$, where $n = 10$ and $r=2$. So $P(10,2)=\frac{10!}{(10 - 2)!}=\frac{10!}{8!}=10\times9 = 90$.

Step2: Calculate ways for second department

For a department of 7 members, with $n = 7$ and $r = 2$, $P(7,2)=\frac{7!}{(7 - 2)!}=\frac{7!}{5!}=7\times6=42$.

Step3: Calculate ways for third department

For a department of 6 members, with $n = 6$ and $r = 2$, $P(6,2)=\frac{6!}{(6 - 2)!}=\frac{6!}{4!}=6\times5 = 30$.

Step4: Use the multiplication principle

Since the selections from each department are independent events, the total number of ways to make all the selections is the product of the number of ways for each department. So the total number of ways is $90\times42\times30$.
$90\times42\times30=(90\times42)\times30 = 3780\times30=113400$.

Answer:

113400