Write the value of _r-1^6 ^56-r C_ 3 + ^50 C_ 4 .

Question

Write the value of $\sum\limits_\text{r-1}^6 {^\text{56-r}}\text{C}_{\text{3}}+{^\text{50}}\text{C}_{\text{4}}.$

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Answer

$\sum\limits_\text{r-1}^6 {^\text{56-r}}\text{C}_{\text{3}}+{^\text{50}}\text{C}_{\text{4}}$
$={^\text{56-1}}\text{C}_{\text{3}}+{^\text{56-2}}\text{C}_{\text{3}}+{^\text{56-3}}\text{C}_{\text{3}}+{^\text{56-4}}\text{C}_{\text{4}}+{^\text{56-5}}\text{C}_{\text{5}}+{^\text{56-6}}\text{C}_{\text{6}}+{^\text{50}}\text{C}_{\text{4}}$
$={^\text{55}}\text{C}_{\text{3}}+{^\text{54}}\text{C}_{\text{3}}+{^\text{53}}\text{C}_{\text{3}}+{^\text{52}}\text{C}_{\text{3}}+{^\text{51}}\text{C}_{\text{3}}+{^\text{50}}\text{C}_{\text{3}}+{^\text{50}}\text{C}_{\text{4}}$
$={^\text{55}}\text{C}_{\text{3}}+{^\text{54}}\text{C}_{\text{3}}+{^\text{53}}\text{C}_{\text{3}}+{^\text{52}}\text{C}_{\text{3}}+{^\text{51}}\text{C}_{\text{3}}+({^\text{50}}\text{C}_{\text{3}}+{^\text{50}}\text{C}_{\text{4}})$
$={^\text{55}}\text{C}_{\text{3}}+{^\text{54}}\text{C}_{\text{3}}+{^\text{53}}\text{C}_{\text{3}}+{^\text{52}}\text{C}_{\text{3}}+({^\text{51}}\text{C}_{\text{3}}+{^\text{51}}\text{C}_{\text{4}})$
$={^\text{55}}\text{C}_{\text{3}}+{^\text{54}}\text{C}_{\text{3}}+({^\text{53}}\text{C}_{\text{3}}+{^\text{54}}\text{C}_{\text{4}})$
$={^\text{55}}\text{C}_{\text{3}}+({^\text{54}}\text{C}_{\text{3}}+{^\text{54}}\text{C}_{\text{4}})$
$=({^\text{55}}\text{C}_{\text{3}}+{^\text{55}}\text{C}_{\text{4}})$
$={^\text{56}}\text{C}_{\text{4}}$

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