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  • Need explanation for: - Binomial theorem and its simple applications - JEE Main-2

For natural numbers m,n if (1-y)^{m}(1+y)^{n}=1+a_{1}y+a_{2}y^{2}+........,\; and\; a_{1}=a_{2}=10,then\; (m,n)\; is

  • Option 1)

    (20, 45)

  • Option 2)

    (35, 20)

  • Option 3)

    (45, 35)

  • Option 4)

    (35, 45).

 

Answers (1)

best_answer

As we learnt in

Expression of Binomial Theorem -

\left ( x+a \right )^{n}= ^{n}\! c_{0}x^{n}a^{0}+^{n}c_{1}x^{n-1}a^{1}+^{n}c_{2}x^{n-2}a^{2}x-----^{n}c_{n}x^{0}a^{n}

 

- wherein

for n  +ve integral .

 

 (1-y)^{m}(1+y)^{n}

\Rightarrow\ \; \left(1-\ ^{m}C_{1}y+\ ^{m}C_{2}y^{2}........ \right ) \left(1+\ ^{n}C_{1}y+\ ^{n}C_{2}y^{2}........ \right )

\Rightarrow\ \; \left(1+(n-m)y+(^{m}C_{2}+^{n}C_{2}-mn)y^{2}.......... \right )

So n - m = 10

and \frac{m(m-1)}{2}+\frac{n(n-1)}{2}-mn=10

Values of (m, n) satisfying are (35, 45)

Correct option is 4.

 


Option 1)

(20, 45)

This is an incorrect option.

Option 2)

(35, 20)

This is an incorrect option.

Option 3)

(45, 35)

This is an incorrect option.

Option 4)

(35, 45).

This is the correct option.

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