3. A club with 8 women and 6 men needs to choose two different members to be president and vice president (combination or permutation).
a. In how many ways is this possible?
b. In how many ways is this possible if women will be chosen as president and a man as vice president?
c. If we select 2 people as volunteers, in how many ways is this possible?
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Sure i can show my work.
I did a and already. answer: total club member is 14 so for a 14.13
and for b 8.6 (need to do the calculation).
But it is not ringing the bell for c. So, if you could please help me or guide me through to do this it will be appreciated. Thank you.
a. To answer this question, we need to use combinations since the order in which the members are chosen does not matter. To find the number of ways to choose 2 members out of 14 (8 women + 6 men), we can use the combination formula:
C(n, r) = n! / (r!(n-r)!)
In this case, we have 14 members to choose from and we want to choose 2 of them, so n = 14 and r = 2. Let's calculate it:
C(14, 2) = 14! / (2!(14-2)!)
= 14! / (2!12!)
= (14 * 13 * 12!) / (2! * 12!)
= (14 * 13) / (2 * 1)
= 7 * 13
= 91
Therefore, there are 91 ways to choose two different members to be president and vice president in this club.
b. In this case, we have 8 women to choose from for the president position and 6 men to choose from for the vice president position. Since the positions are specific (woman as president and man as vice president), this would be considered a permutation.
To find the number of ways, we multiply the number of choices for each position:
Number of ways = Number of choices for president * Number of choices for vice president
= 8 * 6
= 48
Therefore, there are 48 ways to choose a woman as president and a man as vice president in this club.
c. If we need to select 2 people as volunteers without any specific conditions, this question also involves combinations, similar to part a. Since we have a total of 14 members to choose from, we can use the combination formula:
C(n, r) = n! / (r!(n-r)!)
In this case, we want to select 2 people from a group of 14 members, so n = 14 and r = 2. Let's calculate it:
C(14, 2) = 14! / (2!(14-2)!)
= 14! / (2!12!)
= (14 * 13 * 12!) / (2! * 12!)
= (14 * 13) / (2 * 1)
= 7 * 13
= 91
Therefore, there are 91 ways to select 2 people as volunteers in this club.