a club has 15 members plus their president. a 5 person committee is to be formed.
A) how many committees include the president?
B) how many committees don't include the president?
C) how many committees altogether?
a) so we need 4 more from the 15members
= C(15,4) = 1365
b) no president in selection
= C(15,5) = 3003
c) no restriction, choose 5 from the 16
= C(16,5) = 4368
notice: 1365 + 3003 = 4368
To answer this question, let's break it down step by step.
A) To calculate the number of committees that include the president, we need to select 4 additional members from the remaining 15 members (excluding the president). This can be done using the combination formula.
The formula for selecting r items from a set of n items without regard to the order is given by nCr = n! / (r!(n-r)!), where "!" represents the factorial function.
In our case, we have n = 15 (number of members excluding the president) and r = 4 (additional members to be selected). Therefore, the number of committees that include the president is:
15C4 = 15! / (4!(15-4)!) = 1365
So, there are 1365 committees that include the president.
B) To calculate the number of committees that don't include the president, we need to select 5 members from the remaining 15 members (excluding the president). Again, we can use the combination formula:
15C5 = 15! / (5!(15-5)!) = 3003
Therefore, there are 3003 committees that don't include the president.
C) To calculate the total number of committees, we add the number of committees that include the president (1365) to the number of committees that don't include the president (3003):
Total number of committees = 1365 + 3003 = 4368
Therefore, there are 4368 committees in total.