There are 10 councillors and 13 planning department staff available to serve on a committee to look into re establishing two way traffic on the streets of St. Catharines. If the committee will consist of 3 councillors and either 1 or 2 planning staff, how many different committees could the council choose?
To find the number of different committees that the council could choose, we can use combinations (C(n, r)).
We need to choose 3 councillors from a group of 10 councillors, which can be done in C(10, 3) ways. This equals 10! / [(10-3)! * 3!] = 10! / 7! * 3! = (10 * 9 * 8) / (3 * 2 * 1) = 120.
Now, we need to choose either 1 or 2 planning staff to join the committee. This can be done in two ways: (i) selecting only 1 planning staff from a group of 13 or (ii) selecting 2 planning staff from a group of 13.
For Case (i): Choosing 1 planning staff from 13 can be done in C(13, 1) = 13! / [(13 - 1)! * 1!] = 13.
For Case (ii): Choosing 2 planning staff from 13 can be done in C(13, 2) = 13! / [(13 - 2)! * 2!] = (13 * 12) / (2 * 1) = 78.
Now, to find the total number of committees, we sum the numbers of committees in each case:
Total number of committees = Number of committees in Case(i) + Number of committees in Case(ii)
Total number of committees = 13 + 78
Total number of committees = 91
Therefore, the council could choose from 91 different committees.
To determine the number of different committees the council could choose, we need to consider the number of ways to select the councillors and the number of ways to select the planning staff.
First, let's calculate the number of ways to select 3 councillors from a group of 10. This can be done using the combination formula:
C(n, r) = n! / (r!(n-r)!)
where n is the total number of councillors and r is the number of councillors to be selected.
In this case, n = 10 and r = 3, so the number of ways to select the councillors is:
C(10, 3) = 10! / (3!(10-3)!)
= 10! / (3!7!)
= (10 * 9 * 8) / (3 * 2 * 1)
= 120
Now we need to consider the number of ways to select either 1 or 2 planning staff.
For 1 staff member, there are 13 options available.
For 2 staff members, there are C(13, 2) = 13! / (2!(13-2)!) = 13! / (2!11!) = (13 * 12) / (2 * 1) = 78 options.
Therefore, the total number of different committees the council could choose is:
Number of committees = Number of ways to select councillors * Number of ways to select planning staff
= 120 * (13 + 78)
= 120 * 91
= 10,920
So, there are 10,920 different committees the council could choose.