Evaluate ^ 20 C_ 5 + _r=2^5 ^ 25-x C_4. - Vidyadip

Question

Evaluate $^{20}\text{C}_{5}+\sum\limits_\text{r=2}^5\ ^{25-\text{x}}\text{C}_4.$

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Answer

We have,
$\Rightarrow\ ^{20}\text{C}_{5}+\sum\limits_\text{r=2}^5\ ^{25-\text{x}}\text{C}_4$
$\Rightarrow\ \big(^{20}\text{C}_{5}+{^{20}\text{C}_{4}}\big)+{^{21}\text{C}_{4}+^{22}}\text{C}_{4}+{^{23}\text{C}_{4}}$
$\Rightarrow\ \big(^{21}\text{C}_{5}+{^{21}\text{C}_{4}}\big)+{^{22}\text{C}_{4}}+{^{23}\text{C}_{4}}$
$\Rightarrow{^\text{23}}\text{C}_{\text{5}}+{^\text{23}}\text{C}_{\text{4}}$
$\Rightarrow{^\text{24}}\text{C}_{\text{5}}$
$\Rightarrow 42504$

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