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.)

1 1011 ways
0

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

Step 1: Choose the president
There are 35 members in total (16 men and 19 women), so there are 35 options for choosing the president.

Step 2: Choose the vice president
Once the president is chosen, we need to choose the vice president. Since one person has been selected as the president, there are now 34 members remaining to choose from. Therefore, there are 34 options for choosing the vice president.

Step 3: Choose the treasurer
Now that we have chosen the president and vice president, there are 33 members remaining to choose from. Therefore, there are 33 options for choosing the treasurer.

Step 4: Choose the secretary
After choosing the president, vice president, and treasurer, there are 32 members left to choose from. Hence, there are 32 options for choosing the secretary.

Step 5: Choose the advisory committee
For the advisory committee, we need to select six people from the remaining members after choosing the president, vice president, treasurer, and secretary. There are 31 members left after selecting the previous positions, so we need to choose 6 members out of the remaining 31. This can be done using the combination formula:

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

where n is the total number of items to choose from and r is the number of items to be chosen.

In this case, n = 31 and r = 6. Plugging these values into the formula, we get:

C(31, 6) = 31! / (6!(31 - 6)!)

Calculating this gives us:
C(31, 6) = 31! / (6!25!) = 31 × 30 × 29 × 28 × 27 × 26 / (6 × 5 × 4 × 3 × 2 × 1) = 31,122

Step 6: Multiply all the options together
Finally, we multiply all the options together to get the total number of ways:

Total ways = 35 × 34 × 33 × 32 × 31,122 = 1,233,014,720

Therefore, the number of ways the club can choose a president, vice president, treasurer, and secretary, along with an advisory committee of six people, is 1,233,014,720.