Q. 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) $\ ^{55}C_{4}$
Explanation
(a)
$\ ^{50}C_{4} + \sum_{r = 1}^{6}{\ ^{56 - r}C}_{3}$
$= \ ^{50}C_{4} + \ ^{55}C_{3} + \ ^{54}C_{3} + \ ^{53}C_{3} + \ ^{52}C_{3}{+ \ }^{51}C_{3} + \ ^{50}C_{3}$
$= \ ^{51}C_{4}{+ \ }^{51}C_{3} + \ ^{52}C_{3} + \ ^{53}C_{3} + \ ^{54}C_{3} + \ ^{55}C_{3}$
$\lbrack\because\ \ ^{n}C_{r} + \ ^{n}C_{r + 1} = \ ^{n + 1}C_{r + 1}\rbrack$
$= \ ^{52}C_{4} + \ ^{52}C_{3} + \ ^{53}C_{3} + \ ^{54}C_{3} +^{55}C_{3}$
$= \ ^{53}C_{4} + \ ^{53}C_{3} + \ ^{54}C_{3} + \ ^{55}C_{3}$
$= \ ^{54}C_{4} + \ ^{54}C_{3} + \ ^{55}C_{3} = \ ^{55}C_{4} + \ ^{55}C_{3} = \ ^{56}C_{4}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.