There are 12 students interested in creating a shoe shining club. There will be a president, a vice-president, 2 shoe shiners, 2 polish selectors, and the rest will work on advertising.
How many ways can these positions be filled?
I did: (12C1)+(12C1)+(12C2)+(12C2)+(12C6) and got 1015 ways as my answer. Is this correct?
No!
The president can be chosen in C(12,1) ways
the VP can be chosen in C(11,1) ways
the 2 shoe-shiners in C(10,2) ways
the 2 polish selectors in C(8,2) ways, leaving C(6,6) for advertising.
multiply these ....
12x11x45x28x1 = 166320
Thank you Reiny! So just for clarification, you reduce n by how many were chosen previously because those people were already selected and you can't chose them anymore, right?
To find the number of ways these positions can be filled, we need to consider the number of ways each position can be filled individually, and then multiply those together.
1. President: There are 12 students who can be chosen as the president. So, there are 12 ways to fill this position.
2. Vice-President: After the president is chosen, there are 11 students left who can be the vice-president. So, there are 11 ways to fill this position.
3. Shoe Shiners: There are 12 students to choose from for the first shoe shiner position. Once the first shoe shiner is chosen, there are 11 students left to choose from for the second shoe shiner position. Since the order of selection doesn't matter, we divide by 2 to avoid overcounting. So, the number of ways to fill the shoe shiners' positions is (12 * 11) / 2 = 66.
4. Polish Selectors: Similarly, there are 12 students to choose from for the first polish selector position, and 11 students left for the second polish selector position, resulting in (12 * 11) / 2 = 66 ways to fill this position.
5. Advertising: The remaining 6 students will work on advertising. Since the order of selection doesn't matter, we need to find the number of ways to choose 6 students from the remaining 6. This can be calculated as (6C6) = 1 way.
Now, to find the total number of ways these positions can be filled, we multiply the number of ways for each position:
Total ways = 12 * 11 * 66 * 66 * 1 = 653,184
So, the correct answer is 653,184.
To determine the number of ways these positions can be filled, you need to consider a few factors:
1. President: You have 12 students, so you can choose 1 of them as the president. This can be done in 12 ways.
2. Vice-president: After selecting the president, you now have 11 remaining students to choose from for the vice-president position. Therefore, you have 11 choices for the vice-president.
3. Shoe shiners: There are 2 positions available for shoe shiners. After selecting the president and vice-president, you have 10 remaining students to choose from for the first shoe shiner position. Once filled, there are 9 remaining students for the second shoe shiner position.
4. Polish selectors: Similarly, there are 2 positions available for polish selectors. After selecting the president, vice-president, and shoe shiners, there are 8 remaining students to choose from for the first polish selector position. Once filled, there are 7 remaining students for the second polish selector position.
5. Advertising: The remaining students (after selecting the president, vice-president, shoe shiners, and polish selectors) will work on advertising. The number of ways to select the remaining students for advertising is given by the combination C(12, remaining students).
Now, let's calculate the total number of ways for these positions to be filled:
12 * 11 * 10 * 9 * (C(8, remaining students) * C(12, remaining students))
= 11,880 * (C(8, remaining students) * C(12, remaining students))
To find the value of the expression, you need to specify the number of remaining students who will work on advertising. In the given question, it is mentioned that the remaining students will be 6. Hence, the expression becomes:
11,880 * (C(8, 6) * C(12, 6))
= 11,880 * (28 * 924)
= 11,880 * 25,872
= 307,957,760
Therefore, there are 307,957,760 ways to fill these positions, not 1,015 as you calculated.