Question

Assertion :${\sum }_{r=0}^{100}{(}^{500-r}{C}_3){+}^{400}{C}_4{=}^{501}{C}_4$ Reason: ${}^n{C}_r$ + ${}^n{C}_{r-1}$ = ${}^{n+1}{C}_r$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let $S={\sum }_{r=0}^{100}\left({}^{500-r}{C}_3\right){+}^{400}{C}_4$

$S={}^{500}{C}_3+{}^{499}{C}_3...........+{}^{400}{C}_3+{}^{400}{C}_4$

We know that ${}^n{C}_r{+}^n{C}_{r-1}={}^{n+1}{C}_r$

So, $S={}^{500}{C}_3+{}^{499}{C}_3+.............+{}^{401}{C}_3+{}^{400}{C}_3+{}^{400}{C}_4$

$\Rightarrow S={}^{500}{C}_3+{}^{499}{C}_3+.............+{}^{402}{C}_3+{}^{401}{C}_3+{}^{401}{C}_4$

$\Rightarrow S{=}^{500}{C}_3{+}^{499}{C}_3+.............+{}^{403}{C}_3{+}^{402}{C}_3{+}^{402}{C}_4$

Continuing in the same way we get

So, $S={}^{500}{C}_3+{}^{500}{C}_4+={}^{501}{C}_4$

Assertion is correct.

$\therefore$ Given reason is the correct explanation of assertion.