A club consists of 16 men and 19 women. In how many ways can they choose a president, vice president, treasurer, and secretary, along with an advisory committee of six people? (Round the answer to five decimal places.)

To find the number of ways to choose a president, vice president, treasurer, and secretary, along with an advisory committee of six people, you can use the concept of combinations.

Step 1: Calculate the number of ways to choose the president, vice president, treasurer, and secretary from the total number of club members (both men and women).

Since there are 16 men and 19 women, the total number of people to choose from is 16 + 19 = 35.

The number of ways to choose the four positions from these 35 people is given by the combination formula:

C(n, r) = n! / (r!(n - r)!),

where n is the total number of items (35 in this case) and r is the number of items to be chosen (4 in this case).

Using the combination formula, the number of ways to choose the four positions can be calculated as:

C(35, 4) = 35! / (4!(35 - 4)!) = 35! / (4!31!) = (35*34*33*32) / (4*3*2*1) = 52,360.

So, there are 52,360 ways to choose the president, vice president, treasurer, and secretary.

Step 2: Calculate the number of ways to choose the advisory committee of six people from the remaining club members (excluding the four positions already chosen).

Since the four positions have already been filled, we need to choose six people for the advisory committee from the remaining members, which is (35 - 4) = 31 people.

The number of ways to choose six people from these 31 remaining people can be calculated using the combination formula:

C(31, 6) = 31! / (6!(31 - 6)!) = 31! / (6!25!) = (31*30*29*28*27*26) / (6*5*4*3*2*1) = 449,638.

So, there are 449,638 ways to choose the advisory committee.

Step 3: Finally, we need to multiply the number of ways to choose the four positions with the number of ways to choose the advisory committee to get the total number of ways to select both groups.

Total number of ways = Number of ways to choose positions * Number of ways to choose the advisory committee
= 52,360 * 449,638
≈ 23,533,493,280.

Therefore, there are approximately 23,533,493,280 ways to choose a president, vice president, treasurer, and secretary, along with an advisory committee of six people.