Question

Give a combinatorial interpretation of the coefficient of \({x^6}\) in the expansion\({\left( {1 + x + {x^2} + {x^3} + \cdots } \right)^n}\). Use this interpretation to find this number.

Short answer

The combinatorial interpretation of the coefficient of \({x^6}\) is \(C(n + 5,6) = \frac{{(n + 5)!}}{{6!(n - 1)!}}\).

Step-by-step solution

Use Permutation and Combination:

Permutation:

No repetition allowed:\(P(n,r) = \frac{{n!}}{{(n - r)!}}\)

Repetition allowed:\({{\bf{n}}^r}\)

Combination:

No repetition allowed:\(C(n,r) = \left( {\begin{array}{*{20}{l}}n\\r\end{array}} \right) = \frac{{n!}}{{r!(n - r)!}}\)

Repetition allowed:\(C(n + r - 1,r) = \frac{{(n + r - 1)!}}{{r!(n - 1)!}}\) with\(n! = n \cdot (n - 1) \cdot \ldots \cdot 2 \cdot 1\).

The expression is given by:

\({\left( {1 + x + {x^2} + {x^3} + \ldots } \right)^n}\)

I will use the interpretation used in exercise 13: The number of different ways to give 6 identical balloons to \(n\) children where each child receives balloons.

\(r = 6\)

Since the sum \(1 + x + {x^2} + {x^3} + \ldots \) contains all possible powers of\(x\), there is no restriction on the number of balloons that each child has to receive.

The number of different ways to give 6 identical balloons to \(n\) children is represented by the given expansion.

The order of the balloons is not important.

Thus by using the combination, repetition of the same children is also allowed:

\(\begin{array}{c}C(n + 6 - 1,6) = C(n + 5,6)\\ = \frac{{(n + 5)!}}{{6!(n - 1)!}}\end{array}\)