NCERT Class XI Mathematics - Binomial Theorem - Solutions
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Question : 29
Total: 36
Find the coefficient of x 5 ( 1 + 2 x ) 6 ( 1 – x ) 7
Solution:
We first expand each of the factors of the given product using binomial theorem. We have
( 1 + 2 x ) 6
C 0 +
C 1 ( 2 x ) +
C 2 ( 2 x ) 2
C 3 ( 2 x ) 3 +
C 4 ( 2 x ) 4 +
C 5 ( 2 x ) 5
C 6 ( 2 x ) 6
= 1 + 12x +60 x 2 + 160 x 3 240 x 3 + 240 x 4 + 192 x 5 + 64 x 6
and( 1 − x ) 7
C 0 +
C 1 ( − x ) +
C 2 ( − x ) 2
C 3 ( − x ) 3 +
C 4 ( − x ) 4
C 5 ( − x ) 5 +
C 6 ( − x ) 6 +
C 7 ( − x ) 7
= 1 - 7x +21 x 2 − 35 x 3 + 35 x 4 − 21 x 5 7 x 6 − x 7
Thus( 1 + 2 x ) 6 ( 1 − x ) 7
= (1 + 12x +60 x 2 + 160 x 3 + 240 x 3 240 x 4 + 192 x 5 + 64 x 6 21 x 2 − 35 x 3 + 35 x 4 − 21 x 5 7 x 6 − x 7
We write only those terms which involvesx 5
thatx r . x 5 – r x 5 x 5
1( − 21 x 5 ) + 12 x ( 35 x 4 ) 60 x 2 ( − 35 x 3 ) + 160 x 3 ( 21 x 2 ) 240 x 4 ( − 7 x ) + 192 x 5 ( 1 )
=− 21 x 5 + 420 x 5 − 2100 x 5 3360 x 5 − 1680 x 5 + 192 x 5 171 x 5
Thus the coefficients of x5 in the given product is 171.
= 1 + 12x +
and
= 1 - 7x +
Thus
= (1 + 12x +
We write only those terms which involves
that
1
=
Thus the coefficients of x5 in the given product is 171.
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