Question

The co-efficient of $x^5$ in the expansion of ${\left(1+x\right)}^{21}+{\left(1+x\right)}^{22}+........+{\left(1+x\right)}^{30}$ is:

• A. ${}^{51}{C}_5$
• B. ${}^9{C}_5$
• C. ${}^{31}{C}_6{-}^{21}{C}_6$
• D. ${}^{30}{C}_5{+}^{20}{C}_5$

Answer 1

The correct option is C ${}^{31}{C}_6{-}^{21}{C}_6$

According to question,
$(1+x{)}^{21}$+$(1+x{)}^{22}$+..................+$(1+x{)}^{30}$
For , coeff. of $x^5$ in the expansion, we need to take out coeff. of $x^5$ from every terms of given series.
i.e; ${}^{21}{C}_5{+}^{22}{C}_5+.................{+}^{30}{C}_5$
Add & subtract ${}^{21}{C}_6$ in this,
${}^{21}{C}_6{+}^{21}{C}_5{+}^{22}{C}_5+.................{+}^{30}{C}_5{-}^{21}{C}_6$
Since, (${}^n{C}_r{+}^{n}Cr-1{=}^{n+1}{C}_r$)
Therefore,
=> ${}^{22}{C}_6{+}^{22}{C}_5+.................{+}^{30}{C}_5{-}^{21}{C}_6$
=> ${}^{23}{C}_6{+}^{23}{C}_5+.................{+}^{30}{C}_5{-}^{21}{C}_6$
Similarly , we get
=> ${}^{30}{C}_6{+}^{30}{C}_5{-}^{21}{C}_6$
=> ${}^{31}{C}_6{-}^{21}{C}_6$