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 total number of ways the investor can choose mutual funds according to the given criteria, we need to apply the concept of combinations.

First, let's calculate the number of ways the investor can choose from the high-risk category. Since there are 8 mutual funds in the high-risk category, and the investor wants to invest in 3 of them, we use the combination formula:

C(n, r) = n! / (r!(n-r)!)

Where:
n = total number of items (mutual funds)
r = number of items to be chosen (mutual funds to invest in)

Plugging in the numbers, we have:

C(8, 3) = 8! / (3!(8-3)!)
= 8! / (3!5!)

Since factorials can be simplified, we have:

C(8, 3) = (8 * 7 * 6 * 5!)/(3! * 5!)
= (8 * 7 * 6)/(3 * 2 * 1)
= 8 * 7 * 2
= 112

Therefore, there are 112 ways the investor can choose 3 mutual funds from the high-risk category.

Similarly, let's calculate the number of ways the investor can choose from the moderate-risk category:

C(7, 4) = 7! / (4!(7-4)!)
= (7 * 6 * 5!)/(4! * 3!)
= (7 * 6)/(3 * 2 * 1)
= 7 * 2
= 14

Therefore, there are 14 ways the investor can choose 4 mutual funds from the moderate-risk category.

Finally, let's calculate the number of ways the investor can choose from the low-risk category:

C(10, 10) = 10! / (10!(10-10)!)
= 1

Since the investor wants to invest in all 10 mutual funds from the low-risk category, there is only 1 way to do this.

To find the total number of ways the investor can choose mutual funds according to the given criteria, we need to multiply the three results together:

Total number of ways = Number of ways to choose high-risk funds * Number of ways to choose moderate-risk funds * Number of ways to choose low-risk funds

Total number of ways = 112 * 14 * 1
= 1,568

Therefore, the investor can choose mutual funds in 1,568 different ways by investing in 3 high-risk, 4 moderate-risk, and 10 low-risk funds.