Question

Coefficient of $x^6$ in ${\left(1+x\right)}^6+{\left(1+x\right)}^7+....+{\left(1+x\right)}^{15}$ is

• A. ${}^{16}{C}_{10}$
• B. ${}^{16}{C}_8$
• C. ${}^{16}{C}_9$
• D. ${}^{16}{C}_{12}$

Answer 1

The correct option is C ${}^{16}{C}_9$

Coeff of $x^6$ in $a\left)\right(1+x{)}^6$ is ${}^6{C}_6=1$

$b\left)\right(1+x{)}^7$ is ${}^7{C}_6$

$(1+x{)}^{15}$ is ${}^{15}{C}_6$.

$\Rightarrow$ Coefficient of $x^6$ is $(1+x{)}^6+(1+x{)}^7+-----(1+x{)}^{15}$

$=1{+}^7{C}_6{+}^8{C}_6+------{+}^{15}{C}_6$

${=}^7{C}_7{+}^7{C}_6{+}^8{C}_6+-----{+}^{15}{C}_6$.

${=}^8{C}_7{+}^8{C}_6{+}^9{C}_6-----{+}^{15}{C}_6$.

${=}^9{C}_7{+}^9{C}_6+----[{\because }^n{C}_r+n_{r-1}{=}^{n+1}{C}_r]$.

${=}^{16}{C}_7$.

${=}^{16}{C}_9$