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 invest in different risk categories, we can use the concept of combinations.
Combinations help us find the number of ways to choose objects from a set without considering the order in which they are chosen.
In this case, the investor wants to invest in 3 high-risk funds, 4 moderate-risk funds, and 10 low-risk funds. While the specific order in which the funds are chosen doesn't matter, the category each fund falls into is important.
The formula to find the number of combinations is:
nCr = n! / (r!(n-r)!),
where n is the total number of options available and r is the number of choices we want to make.
For the high-risk funds, there are 8 options and the investor wants to choose 3. So, we can calculate:
C(8, 3) = 8! / (3!(8-3)!) = 8! / (3!5!) = (8 * 7 * 6) / (3 * 2 * 1) = 56.
Similarly, for the moderate-risk funds:
C(7, 4) = 7! / (4!(7-4)!) = 7! / (4!3!) = (7 * 6 * 5) / (3 * 2 * 1) = 35.
Lastly, for the low-risk funds:
C(10, 10) = 10! / (10!(10-10)!) = 10! / (10!0!) = 1.
To find the total number of ways the investor can invest, we multiply the combinations for each risk category:
56 * 35 * 1 = 1960.
Therefore, there are 1960 ways the investor can allocate their investments among the different risk categories.