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Binomial Theorem — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsBinomial Theorem1 Mark
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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