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)

${}^{56}{C}_4$

We need to find value of ${}^{50}{C}_4+{\sum }_{r=1}^6{}^{56-r}{C}_3$

${=}^{50}{C}_4{+}^{50}{C}_3{+}^{51}{C}_3{+}^{52}{C}_3{+}^{53}{C}_3{+}^{54}{C}_3{+}^{55}{C}_3$

$\left({\because }^n{C}_i{+}^n{C}_{i-1}{=}^{n+1}{C}_i\right)$

${=}^{51}{C}_4{+}^{51}{C}_3{+}^{52}{C}_3{+}^{53}{C}_3{+}^{54}{C}_3{+}^{55}{C}_3$

${=}^{52}{C}_4{+}^{52}{C}_3{+}^{53}{C}_3{+}^{54}{C}_3{+}^{55}{C}_3$

${=}^{53}{C}_4{+}^{53}{C}_3{+}^{54}{C}_3{+}^{55}{C}_3$

${=}^{54}{C}_4{+}^{54}{C}_3{+}^{55}{C}_3$

${=}^{55}{C}_4{+}^{55}{C}_3$

${=}^{56}{C}_4$