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Help me please, - Binomial theorem and its simple applications - JEE Main

The coefficient of x^{10} in the expansion of \left ( 1+x \right )^{2}\left ( 1+x^{2} \right )^{3}\left ( 1+x^{3} \right )^{4} is equal to :

  • Option 1)

    52

  • Option 2)

    56

  • Option 3)

    50

  • Option 4)

    44

 
Answers (1)
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As we learned

 

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 .

 

 

The expresion is 

\left ( 1+x^{2} \right )^{2}\left ( 1+x^{2} \right )^{3}\left ( 1+x^{2} \right )^{4}

\Rightarrow \: \left ( 1+2x+x^{2} \right )\left ( 1+x^{2} \right )^{3}\left ( 1+x^{3} \right )^{4}

\Rightarrow \: \left ( 1+2x+x^{2} \right )\left ( 1+^{4}C_{1}x^{3}+^{4}C_{2}x^{6}+^{4}C_{3}x^{9}+^{4}C_{4}x^{12} \right )\left ( 1+x^{2} \right )^{3}

Coeff of x^{10}\Rightarrow \: 1\times ^{4}C_{2}\times ^{3}C_{2}+2\times ^{4}C_{1}\times ^{3}C_{3}+2\times ^{4}C_{3}\times 1+1\times ^{4}C_{2}\times ^{3}C_{1}

\Rightarrow \: 18+8+8+18

\Rightarrow \: 52


Option 1)

52

Option 2)

56

Option 3)

50

Option 4)

44

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