Question

Assertion :Let $n,m\in N$ and $n\ge m$ Reason: ${}^n{C}_m+2^{n-1}{C}_m+3^{n-2}{C}_m+...{+}^m{C}_m{=}^{n+1}{C}_{m+1}$

• 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. Both Assertion and Reason are incorrect

Answer 1

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

We can write the expression in statement $-2$ as

${}^m{C}_m{+}^{m+1}{C}_m{+}^{m+2}{C}_m+...{+}^{n-1}{C}_m{+}^m{C}_m$

$=\left[{}^{m+1}{C}_{m+1}{+}^{m+1}{C}_m\right]{+}^{m+2}{C}_m+...{+}^{n-1}{C}_m{+}^n{C}_m$

$[{\because }^m{C}_m=1{=}^{m+1}{C}_{m+1}]$

${=}^{m+2}{C}_{m+1}{+}^{m+2}{C}_m+...{+}^{n-1}{C}_m{+}^n{C}_m$

$\left[{\because }^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r\right]$

${=}^{m+3}{C}_{m+1}{+}^{m+3}{C}_m+...{+}^{n-1}{C}_m{+}^n{C}_m$

$=...$

${=}^n{C}_{m+1}{+}^n{C}_m{=}^{n+1}{C}_{m+1}$

We can write expression in statement $-2$ as follows;

${}^n{C}_m{+}^{n-1}{C}_m{+}^{n-2}{C}_m+...{+}^m{C}_m$

${+}^{n-1}{C}_m{+}^{n-2}{C}_m+...{+}^m{C}_m$

${+}^{n-2}{C}_m+...{+}^m{C}_m$

$+...$

${+}^{m+1}{C}_m{+}^m$

${+}^m{C}_m$

Using the above relation on each row, we find its equal to

${}^{n+1}{C}_{m+1}{+}^{n-1}{C}_{m+1}{+}^{n-2}{C}_{m+1}+...{+}^{m+1}{C}_{m+1}$

${=}^{n+1+}{C}_{m+1+1}{=}^{n+2}{C}_{m+2}$

$[$using the above relation$]$