# NCERT Class 11 Mathematics Solutions: Chapter 8 – Binomial Theorem Miscellaneous Exercise Part 3 (For CBSE, ICSE, IAS, NET, NRA 2023)

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1. Find the coefficient of in the product using binomial theorem.

Answer:

Using Binomial Theorem, the expressions, , can be expanded as

The complete multiplication of the two brackets is not required to be carried out.

Only those terms, which involve , are required.

The terms containing are

The coefficient of in the given product is .

2. If are distinct integers, prove that is a factor of , whenever is a positive integer. [Hint: write and expand]

Answer:

In order to prove that ) is a factor of , it has to be proved that , where is some natural number

It can be written that

Where, is a natural number.

This shows that is a factor of , where n is a positive integer.