Question

The value of ${}^{50}{C}_4+$ $\sum _{r=1}^6$ ${}^{56-r}{C}_3$ is :

• A. ${}^{56}{C}_4$
• B. ${}^{56}{C}_3$
• C. ${}^{55}{C}_3$
• D. ${}^{55}{C}_4$

Answer 1

Answer: (A)

Key Idea : ${}^n{C}_r{+}^n{C}_{r+1}{=}^{n+1}{C}_{r+1}.$
Now, ${}^{50}{C}_4{+}^{55}{C}_3{+}^{54}{C}_3{+}^{53}{C}_3{+}^{52}{C}_3{+}^{51}{C}_3{+}^{50}{C}_3$
${=}^{50}{C}_3{+}^{50}{C}_4{+}^{51}{C}_3{+}^{52}{C}_3{+}^{53}{C}_3{+}^{54}{C}_3{+}^{55}{C}_3$
${=}^{51}{C}_4{+}^{51}{C}_3{+}^{52}{C}_3{+}^{53}{C}_3{+}^{54}{C}_3{+}^{55}{C}_3$
$\left({\because }^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r\right)$
${=}^{52}{C}_4{+}^{52}{C}_3{+}^{53}{C}_3{+}^{54}{C}_3{+}^{55}{C}_3$
${=}^{53}{C}_4{+}^{53}{C}_3{+}^{54}{C}_3{+}^{55}{C}_3$
${=}^{54}{C}_4{+}^{54}{C}_3{+}^{55}{C}_3$
${=}^{55}{C}_4{+}^{55}{C}_3{=}^{56}{C}_4$