The value of C 4 50 + r = 1 6 C 3 56 - r is

Question

The value of ${}^{50}C_{4} + \sum_{r = 1}^{6} {}^{56 - r}C_{3}$ is

Smart Achievers · Accepted Answer

(1)

${}^{50}C_{4} + \sum_{r = 1}^{6} {}^{56 - r}C_{3}$

$Putting r = 6, 5, 4, 3, 2, 1, we get$

${}^{50}C_{4} + {}^{50}C_{3} + {}^{51}C_{3} + {}^{52}C_{3} + {}^{53}C_{3} + {}^{54}C_{3} + {}^{55}C_{3}$ $(∵ {}^{n}C_{r} + {}^{n}C_{r + 1} = {}^{n + 1}C_{r + 1})$

$= {}^{51}C_{4} + {}^{51}C_{3} + {}^{52}C_{3} + {}^{53}C_{3} + {}^{54}C_{3} + {}^{55}C_{3}$

$= {}^{52}C_{4} + {}^{52}C_{3} + {}^{53}C_{3} + {}^{54}C_{3} + {}^{55}C_{3}$

$= {}^{53}C_{4} + {}^{53}C_{3} + {}^{54}C_{3} + {}^{55}C_{3}$

$= {}^{54}C_{4} + {}^{54}C_{3} + {}^{55}C_{3}$

$= {}^{55}C_{4} + {}^{55}C_{3} = {}^{56}C_{4}$

Solution

Q.
The value of ${}^{50}C_{4} + \sum_{r = 1}^{6} {}^{56 - r}C_{3}$ is

1 ${}^{56}C_{4}$

2 ${}^{56}C_{3}$

3 ${}^{55}C_{3}$

4 ${}^{55}C_{4}$

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Solution

Q. The value of C450+∑r=16C356-r is

1 C456

2 C356

3 C355

4 C455