Answer to Question #341167 in Combinatorics | Number Theory for Ana

Question

Question

A 33​-term series is written using only the integers +9 and −2. How many such series can be written that have a sum of​0?

Answer

The number of series that can be written is 1 107 568.

Request and Details

Can you please provide a process for solving this question? It may involve permutations and combinations.

Answer 1

We can form 0, by using 2(+9) and 9(-2).

2x9-9x2=0. This is 11(2 digit "+9" and 9 digit "-2") digits. 33/11=3. So to construct serie whith sum 0, we should distribute 2x3=6 digits "9" by 33 terms. We can do it by $C^6_{33}=\frac{33!}{27!6!}=1107568$ ways.