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The coefficient of x^5 in the expansion of (1+x)^{21}+(1+x)^{2

Question

The coefficient of $x^{5}$ in the expansion of $(1 + x)^{21} + (1 + x)^{22} + ........ + (1 + x)^{30}$ is:

^{51} C_{5}

^{9} C_{5}

^{31} C_{6} -^{21} C_{6}

^{30} C_{5} +^{20} C_{5}

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Isabella Discussed

The coefficient of

x^{5}

in the expansion of

(1 + x)^{21} + (1 + x)^{22} + ........ + (1 + x)^{30}

is:

13 mins ago

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13 mins ago

Text solutionVerified

Given :

(1 + x)^{21} + (1 + x)^{22} + (1 + x)^{23} + ..... + (1 + x)^{30}

Coefficient of

x^{5}

^{21} C_{5} (1)^{16} x^{5} +^{22} C_{5} (1)^{17} x^{5} +^{23} C_{5} (1)^{18} x^{5} + .... +^{29} C_{5} (1)^{24} x^{5} +^{30} C_{5} (1)^{25} x^{5}

= x^{5} [^{21} C_{5} +^{22} C_{5} +^{23} C_{5} + ..... +^{29} C_{5} +^{30} C_{5}] ....... (1)

Now we know

^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}

∴

Adding and substracting

^{21} C_{6}

from eq

(1)

the coefficient of

x^{5}

^{21} C_{5} +^{21} C_{6} +^{22} C_{5} +^{25} C_{5} + ..... +^{29} C_{5} +^{30} C_{5} -^{21} C_{6}

=^{22} C_{6} +^{22} C_{5} +^{23} C_{5} + ..... +^{29} C_{5} +^{30} C_{5} -^{21} C_{6}

=^{23} C_{6} +^{23} C_{5} + .... +^{29} C_{5} +^{30} C_{5} -^{21} C_{6}

This way

^{30} C_{6} +^{30} C_{5} -^{21} C_{6}

\Rightarrow^{31} C_{6} -^{21} C_{6}

[Using

^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}

]

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Question Text	The coefficient of $x^{5}$ in the expansion of $(1 + x)^{21} + (1 + x)^{22} + ........ + (1 + x)^{30}$ is:
Answer Type	Text solution:1
Upvotes	151