The coefficient of $x^{6}$ in $(1 + x)^{6}$ + $(1 + x)^{7}$ ......+ $(1 + x)^{15}$ is

^{16} C_{9}

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^{16} C_{5} -^{6} C_{5}

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^{16} C_{6} - 1

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none of these

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Solution

The correct option is A $^{16} C_{9}$
Coefficient of $x^{6}$ is
$^{6} C_{6} +^{7} C_{6} +^{8} C_{6} + . . . .^{15} C_{6}$
Now
$^{n} C_{r} +^{n + 1} C_{r} =^{n + 1} C_{r + 1}$ ...(i)
$[^{6} C_{6} -^{7} C_{7}] +^{7} C_{7} +^{7} C_{6} +^{8} C_{6} + . . . .^{15} C_{6}$
$= 0 +^{7} C_{7} +^{7} C_{6} +^{8} C_{6} + . . . .^{15} C_{6}$
$=^{8} C_{7} +^{8} C_{6} +^{9} C_{6} + . . . .^{15} C_{6}$ using i
$=^{9} C_{7} +^{9} C_{6} +^{10} C_{6} + . . . .^{15} C_{6}$
:
:
$=^{15} C_{7} +^{15} C_{6}$
$=^{16} C_{7}$
$=^{16} C_{9}$

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The coefficient of x6 in (1+x)6+(1+x)7......+(1+x)15 is

The correct option is A 16C9Coefficient of x6 is 6C6+7C6+8C6+....15C6Now nCr+n+1Cr=n+1Cr+1 ...(i)[6C6−7C7]+7C7+7C6+8C6+....15C6=0+7C7+7C6+8C6+....15C6=8C7+8C6+9C6+....15C6 using i=9C7+9C6+10C6+....15C6::=15C7+15C6=16C7=16C9

The coefficient of $x^{6}$ in $(1 + x)^{6}$ + $(1 + x)^{7}$ ......+ $(1 + x)^{15}$ is