An advisor offers 8 mutual funds in the high risk category, 7 moderate, 10 in low risk.
The investor decides to invest in 3 high risk, 4 moderate risk, and 10 low risk. How many ways can he do this?
To find the number of ways the investor can choose the mutual funds, we need to calculate the combinations of the high, moderate, and low-risk categories.
The investor wants to select 3 mutual funds from the high-risk category. We can calculate this using the combination formula:
C(n, r) = n! / (r!(n-r)!)
In this case, n = 8 (the number of mutual funds in the high-risk category) and r = 3 (the number of mutual funds the investor wants to select). Plugging these values into the formula:
C(8, 3) = 8! / (3!(8-3)!)
= 8! / (3!5!)
= (8 * 7 * 6) / (3 * 2 * 1)
= 56
So, there are 56 ways for the investor to choose 3 mutual funds from the high-risk category.
Similarly, for the moderate-risk category, the investor wants to select 4 mutual funds out of 7. Calculating the combination:
C(7, 4) = 7! / (4!(7-4)!)
= 7! / (4!3!)
= (7 * 6 * 5) / (3 * 2 * 1)
= 35
Therefore, there are 35 ways for the investor to choose 4 mutual funds from the moderate-risk category.
Lastly, for the low-risk category, the investor wants to select 10 mutual funds out of the 10 available options, which means selecting all of them. In this case, there is only one way to do so.
Now, to find the total number of ways the investor can select the mutual funds, we need to multiply the number of ways for each category:
Total number of ways = Number of ways for high-risk * Number of ways for moderate-risk * Number of ways for low-risk
= 56 * 35 * 1
= 1960
Therefore, there are 1960 ways for the investor to invest in 3 high-risk, 4 moderate-risk, and 10 low-risk mutual funds.