20 million is to be invested in 4 companies A, B, C, D. The minimum amount for investments are 1, 2, 3, 4 million respectively. How many different investment strategies are available if
- An investment has to be made in each company?
- An investment has to be made in at-least 3 of the 4 companies?
For first part I distribute the minimum required money for investment in 1 way. Now I have 10 million remaining to invest in 4 companies. I do this using bars and stars in 13C3 ways.
For second part we can invest in 3 or 4 companies. For 4 companies the calculation of first part would suffice. But for 3 companies , my approach seems to be a bit messier:
First I choose 3 companies in which to invest in 4C3 ways.
For each combination , I first invest the minimum required amount for the three companies in 1 way and then use bars and stars to distribute the remaining money in these 3 companies.
Finally I add all the ways.
My questions are :
- Is my approach for both parts correct?
- For part 2 is there some alternative approach , that is less messy and requires less computation? My approach will become unbearable once the number of companies increase.
I would have solved it exactly the same way as you did.
For the second part let $N_a,N_b,N_c,N_d$ be the number of investements that invest in all companies except $a,b,c,d$, $N_a$ is $\binom{13}{2}$ since we have $11$ "free" millions and $2$ bars. From here find $N_a+N_b+N_c+N_d=\binom{13}{2}+\binom{14}{2}+\binom{15}{2}+\binom{16}{2}$. I really don't think it is that messy.