Find the value of\(^{50}C_4 + \displaystyle\sum_{r=1}^{6} \space^{56-r

1.	Find the value of\(^{50}C_4 + \displaystyle\sum_{r=1}^{6} \space^{56-r}C_3\)
Answer» Given expression is \(^{50}C_4 + \displaystyle\sum_{r=1}^{6} \space^{56-r}C_3\) \(=\space^{56}C_4+^{55}C_3+^{54}C_3+^{53}C_3+^{52}C_3+^{51}C_3+^{50}C_3\) Writing the terms in reverse order, we get =\((^{50}C_4 +^{50}C_3)+^{51}C_3+^{52}C_3+^{53}C_3+^{54}C_3+^{55}C_3\)= \(^{50}C_4+^{51}C_3+^{52}C_3+^{53}C_3+^{54}C_3+^{55}C_3\) [∴ \(^nC_r+^nC_{r-1}=^{n+1}C_r\)] = \((^{51}C_4+^{51}C_3)+^{52}C_3+^{53}C_3+^{54}C_3+^{55}C_3\) = \((^{52}C_4+^{52}C_3)+^{53}C_3+^{54}C_3+^{55}C_3\) = \((^{53}C_4+^{53}C_3)+^{54}C_3+^{55}C_3\) = \((^{54}C_4+^{54}C_3)+^{55}C_3\) = \(^{55}C_4+^{55}C_3\) = \(^{56}C_4\)

Find the value of\(^{50}C_4 + \displaystyle\sum_{r=1}^{6} \space^{56-r}C_3\)

Answer»

Given expression is

\(^{50}C_4 + \displaystyle\sum_{r=1}^{6} \space^{56-r}C_3\)

\(=\space^{56}C_4+^{55}C_3+^{54}C_3+^{53}C_3+^{52}C_3+^{51}C_3+^{50}C_3\)

Writing the terms in reverse order, we get

=\((^{50}C_4 +^{50}C_3)+^{51}C_3+^{52}C_3+^{53}C_3+^{54}C_3+^{55}C_3\)= \(^{50}C_4+^{51}C_3+^{52}C_3+^{53}C_3+^{54}C_3+^{55}C_3\)

[∴ \(^nC_r+^nC_{r-1}=^{n+1}C_r\)]

= \((^{51}C_4+^{51}C_3)+^{52}C_3+^{53}C_3+^{54}C_3+^{55}C_3\)

= \((^{52}C_4+^{52}C_3)+^{53}C_3+^{54}C_3+^{55}C_3\)

= \((^{53}C_4+^{53}C_3)+^{54}C_3+^{55}C_3\)

= \((^{54}C_4+^{54}C_3)+^{55}C_3\)

= \(^{55}C_4+^{55}C_3\)

= \(^{56}C_4\)