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7.Binomial Theorem
hard
If for some positive integer $n,$ the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in the ratio $5: 10: 14,$ then the largest coefficient in this expansion is
A
$792$
B
$252$
C
$462$
D
$330$
(JEE MAIN-2020)
Solution
Let $n+5=N$
$N _{ C _{ r -1}}: N _{ C _{ r }}: N _{ C _{ r +1}}=5: 10: 14$
$\Rightarrow \frac{ N _{ C _{r}}}{ N _{ C _{ r -1}}}=\frac{ N +1- r }{ r }=2$
$\frac{N_{C_{r+1}}}{N_{C_{r}}}=\frac{N-r}{r+1}=\frac{7}{5}$
$\Rightarrow \quad r=4, N=11$
$\Rightarrow \quad(1+x)^{11}$
Largest coefficient $={ }^{11} C _{6}=462$
Standard 11
Mathematics
