Question

How many different terms are there in the expansion of \({\left( {{{\rm{x}}_{\rm{1}}}{\rm{ + }}{{\rm{x}}_{\rm{2}}}{\rm{ + }}......{\rm{ + }}{{\rm{x}}_{\rm{m}}}} \right)^{\rm{n}}}\)after all terms with identical sets of exponents are added?

Short answer

Number of different terms in the expansion of:

\({\left( {{{\rm{x}}_{\rm{1}}}{\rm{ + }}{{\rm{x}}_{\rm{2}}}{\rm{ + \ldots \ldots }}{\rm{. + xm}}} \right)^{\rm{n}}}{\rm{ = n + m - }}{{\rm{1}}_{{{\rm{C}}_{\rm{n}}}}}\)

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.

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Considering the given information:

\({\left( {{{\rm{x}}_{\rm{1}}}{\rm{ + }}{{\rm{x}}_{\rm{2}}}{\rm{ + \ldots \ldots + xm}}} \right)^{\rm{n}}}\quad {\rm{ }}\)

Using the following concept:

Number of different terms in the expansion \({\left( {{{\rm{x}}_{\rm{1}}}{\rm{ + }}{{\rm{x}}_{\rm{n}}}} \right)^{\rm{n}}}{\rm{ = n + 1}}\)

There are a number of different terms in the expansion of

\(\begin{aligned}{c}{\left( {{{\rm{x}}_{\rm{1}}}{\rm{ + }}{{\rm{x}}_{\rm{2}}}} \right)^{\rm{n}}}{\rm{ &= n + 1}}\\{\rm{ &= n + }}{{\rm{1}}_{{{\rm{C}}_{\rm{1}}}}}\\{\rm{ &= n + 2 - }}{{\rm{1}}_{{{\rm{C}}_{{\rm{2 - 1}}}}}}\end{aligned}\)

So, comparing above,

The number of different terms in the expansion

\({\left( {{{\rm{x}}_{\rm{1}}}{\rm{ + }}{{\rm{x}}_{\rm{2}}}{\rm{ + \ldots \ldots + xm}}} \right)^{\rm{n}}}{\rm{ = n + m - }}{{\rm{1}}_{{{\rm{c}}_{{\rm{m - 1}}}}}}\)

Since, \({{\rm{n}}_{{{\rm{C}}_{\rm{r}}}}}{\rm{ = }}{{\rm{n}}_{{{\rm{C}}_{{\rm{k - }}}}}}\)

\({\rm{SO}}\)

\({\rm{n + m - }}{{\rm{1}}_{{{\rm{C}}_{{\rm{m - 1}}}}}}{\rm{ = n + m - }}{{\rm{1}}_{{{\rm{C}}_{\rm{n}}}}}\)

Therefore, the required number of different terms:

\({\left( {{{\rm{x}}_{\rm{1}}}{\rm{ + }}{{\rm{x}}_{\rm{2}}}{\rm{ + \ldots \ldots }}{\rm{. + xm}}} \right)^{\rm{n}}}{\rm{ = n + m - }}{{\rm{1}}_{{{\rm{C}}_{\rm{n}}}}}\)