Question

How many terms are there in the expansion of \({{\rm{(x + y + z)}}^{{\rm{100}}}}{\rm{?}}\)

Short answer

The number of terms in the expansion of \({{\rm{(x + y + z)}}^{{\rm{100}}}}{\rm{ = 5151}}\).

Step-by-step solution

Definition of Concept

String: A Number String is a collection of related math problems designed to teach number-based strategies. It is a 10- to 15-minute routine that can be used during math class. Gather students in an area where the entire class meets to prepare an area for using the number string.

Find the number of terms in the expansion

Considering the given information:

\({{\rm{(x + y + z)}}^{{\rm{100}}}}\)

Using the following concept:

The number of terms in the \({{\rm{(x + y + z)}}^{\rm{n}}}\) expansion is\(\frac{{{\rm{(n + 2)(n + 1)}}}}{{\rm{2}}}\).

The number of terms included in the expansion of

\(\begin{array}{c}{{\rm{(x + y + z)}}^{{\rm{100}}}}{\rm{ = }}\frac{{{\rm{(100 + 2)(100 + 1)}}}}{{\rm{2}}}\\{\rm{ = }}\frac{{{\rm{102 \times 101}}}}{{\rm{2}}}\\{\rm{ = 51 \times 101}}\\{\rm{ = 5151}}\end{array}\)

Therefore, the required number of terms in the expansion of\({{\rm{(x + y + z)}}^{{\rm{100}}}}\)is \({\rm{5151}}\).