Question

Find the coefficient of \({{\rm{x}}^{\rm{3}}}{{\rm{y}}^{\rm{2}}}{{\rm{z}}^{\rm{5}}}\) in\({{\rm{(x + y + z)}}^{{\rm{10}}}}\).

Short answer

The co-efficient of \({{\rm{x}}^{\rm{3}}}{{\rm{y}}^{\rm{2}}}{{\rm{z}}^{\rm{5}}}\) in \({{\rm{(x + y + z)}}^{{\rm{10}}}}\) is 2520.

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 coefficient

Considering the given information:

\({{\rm{(x + y + z)}}^{{\rm{10}}}}\)

Using the following concept:

The co-efficient of \({{\rm{x}}^{\rm{p}}}{{\rm{y}}^{\rm{q}}}{{\rm{z}}^{\rm{r}}}\) in \({{\rm{(x + y + z)}}^{\rm{n}}}\) is \(\frac{{{\rm{n!}}}}{{{\rm{p!q!r!}}}}\).

Co-efficient of \({{\rm{x}}^{\rm{3}}}{{\rm{y}}^{\rm{2}}}{{\rm{z}}^{\rm{5}}}\) in \({{\rm{(x + y + z)}}^{{\rm{10}}}}\) is

\(\begin{array}{l}{\rm{ = }}\frac{{{\rm{10!}}}}{{{\rm{3!2!5!}}}}\\{\rm{ = }}\frac{{{\rm{10 \times 9 \times 8 \times 7 \times 6 \times 5!}}}}{{{\rm{3 \times 2 \times 2 \times 5!}}}}\\{\rm{ = 2520}}\end{array}\)

Therefore, the required co-efficient of \({{\rm{x}}^{\rm{3}}}{{\rm{y}}^{\rm{2}}}{{\rm{z}}^{\rm{5}}}\) in \({{\rm{(x + y + z)}}^{{\rm{10}}}}\) is 2520.