To study the unity of Canada, a seven=person committee is to be selected from five public figures from Western Canada, five public figures from Central Canada, and five public figures from Eastern Canada.

I figured out parts a & b to this question, but I'm having troubles with c.

How many committees would have at least one representative from each region of Canada?

suppose nobody from WC is chosen

no of ways to choose the committee is (10,6) or 210
The same would be true for each of the other two regions.
BTW, there is no way that 2 regions can be excluded, since we need 7 people and there would be only 5 available.

So what we don't want is 3(210) or 630 , where one of the regions in excluded.

without restrictions, no of committee's is (15,7) = 6435

So number of committees with at least one person from each region is 6435 - 630 = 5805

;S

i checked the answer key
and it says 6075

To determine the number of committees that would have at least one representative from each region of Canada, we can use the principle of inclusion-exclusion.

Step 1: Calculate the total number of possible committees without any restrictions.
Since each region has 5 public figures, we can choose one representative from each region independently. Therefore, the total number of possible committees without any restrictions is:
(5 choices from Western Canada) * (5 choices from Central Canada) * (5 choices from Eastern Canada) = 5^3 = 125.

Step 2: Calculate the number of committees with no representative from at least one region.
To find the number of committees without a representative from Western Canada, we need to consider only the other two regions: Central Canada and Eastern Canada. Since there are 5 public figures from each of these regions, the number of committees without any representative from Western Canada is:
(5 choices from Central Canada) * (5 choices from Eastern Canada) = 5^2 = 25.

Since the same reasoning applies to the other two regions, the number of committees without a representative from Central Canada and Eastern Canada is also 25.

Step 3: Calculate the number of committees without any representatives from at least two regions.
To find the number of committees without any representative from Western Canada and Central Canada, we need to consider only Eastern Canada. Since there are 5 public figures from Eastern Canada, the number of committees without any representative from Western Canada and Central Canada is:
(5 choices from Eastern Canada) = 5.

Again, the same reasoning applies to the other combinations of regions. So, the number of committees without any representative from Western Canada and Central Canada, as well as Eastern Canada and either of the other two regions, is also 5.

Step 4: Apply the principle of inclusion-exclusion.
To calculate the number of committees that have at least one representative from each region, we need to subtract the number of committees without any representative from at least one region from the total number of committees.

Total number of committees with at least one representative from each region = Total number of possible committees without any restrictions - Number of committees with no representative from at least one region.
= 125 - (25 + 25 + 25 - 5 - 5 - 5)
= 125 - 65
= 60.

Therefore, there would be 60 committees that have at least one representative from each region of Canada.