Number of combinations and permutations of length $5$ obtained from "Mississippi"

Question

Please help me to solve this: How many permutation & combination can be be formed from the word $MISSISSIPPI$ taking $5$ at a time?

Answer 1

The number of combinations is the number of solutions of $$m+i+s+p=5,\quad0\le m\le1,\quad0\le i\le4,\quad 0\le s\le4,\quad0\le p\le2$$ where $m$ stands for the number of time you use $M$, and so on. This is a standard kind of "Hpw many solutions to $a+b+\cdots+k=n$ with restrictions on the unknowns" kind of problem that has been asked and answered on this website many times.

The permutations problem is messier. Give it to a computer.

Answer 2

For the number of permutations:

The generating function for your problem is:

$$(1+x) \left(1+x+\frac{x^2}{2}\right) \left(1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}\right)^2$$

when we expand this we look at the coefficient of $x^5$ which is $ 5! \left(\frac{55x^5}{12}\right) $

This becomes 550 when 1 is substituted for x and that is the answer.

For the combinations where order does not matter, this instead uses the generating function:

$$(1+x)(1+x+x^2)(1+x+x^2+x^3+x^4)^2$$

Generating functions Link