JEE MAIN - Mathematics (2020 - 7th January Evening Slot - No. 2)
The coefficient of x7
in the expression
(1 + x)10 + x(1 + x)9 + x2(1 + x)8 + ......+ x10 is:
(1 + x)10 + x(1 + x)9 + x2(1 + x)8 + ......+ x10 is:
120
330
420
210
Explanation
(1 + x)10 + x(1 + x)9
+ x2(1 + x)8
+ ......+ x10
This is a G.P where
First term, a = (1 + x)10
common ratio, r = $${x \over {1 + x}}$$
Number of terms = 11
Sum of G.P
= $${{{{\left( {1 + x} \right)}^{10}}\left( {1 - {{\left( {{x \over {1 + x}}} \right)}^{11}}} \right)} \over {1 - {x \over {1 + x}}}}$$
= (1 + x)11 – x11
So Coefficient of x7 is 11C7 = 330
This is a G.P where
First term, a = (1 + x)10
common ratio, r = $${x \over {1 + x}}$$
Number of terms = 11
Sum of G.P
= $${{{{\left( {1 + x} \right)}^{10}}\left( {1 - {{\left( {{x \over {1 + x}}} \right)}^{11}}} \right)} \over {1 - {x \over {1 + x}}}}$$
= (1 + x)11 – x11
So Coefficient of x7 is 11C7 = 330
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