Combinatorics: Cookies on a Line 2

You have 7 indistinguishable oatmeal cookies and 8 indistinguishable peanut butter cookies. In how many ways can you arrange 9 of these cookies in a straight line?

Answer

Let $$i$$ be the number of oatmeal cookies in the arrangement. Since you can use at most 7 oatmeal cookies and you need 9 cookies in total, the number of peanut butter cookies will be $$9-i$$ , which must be at most 8. This gives the constraint:

9-i \le 8 \quad \Longrightarrow \quad i \ge 1.

Thus, $$i$$ can range from 1 to 7. For each valid $$i$$ , the number of arrangements is given by choosing $$i$$ positions (out of 9) for the oatmeal cookies:

\binom{9}{i}.

The total number of arrangements is

\sum_{i=1}^{7} \binom{9}{i}.

This number is $$501$$ .

Notes and comments

Comment 1: This differs from the "Cookies on a Line 1" problem because here we must have at least one cookie from each type (we cannot form the line only with one type) to make a total of 9 cookies (due to the limited availability), which imposes the constraint $1 \le i \le 7$ .