How many 10-character strings can be made having only 4 letters of 'a', 'b', 'c' and 'd' (16 in total)

Question

I know how many there are 16-char strings $$ \frac{16!}{4!4!4!4!} $$

How to avoid using multiple case scenario with above technique? Is there a generic solution to this kind of problem?

There is no easier way to do it, but you can simplify your work by taking the 16-character string and figuring out how many ways you can REMOVE 6 characters from it and then taking into account inclusion/exclusion. — snowfall512, Aug 09 '16 at 12:03

score 2 · Accepted Answer · edited Apr 13 '17 at 12:21

2

There is a formula based on generating functions.

coefficient of $x^{10}$ in $10!(1+x +\frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!})^4 = 722,400$

See this answer on MSE for why it works.

edited Apr 13 '17 at 12:21

Community

answered Aug 09 '16 at 12:47

true blue anil

1 Answers1