If for $3 \leq r \leq 30$,$$\left( \ ^{30}C_{30 - r} \right) + 3\left( \ ^{30}C_{31 - r} \right) + 3

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If for $3 \leq r \leq 30$,$$\left( \ ^{30}C_{30 - r} \right) + 3\left( \ ^{30}C_{31 - r} \right) + 3\left( \ ^{30}C_{32 - r} \right) + \left( \ ^{30}C_{33 - r} \right) = \ ^{m}C_{r},$$then $m$ equals:

Accepted Answer

Sol. [Key Concept: Combine by Pascal's rule: LHS $= {}^{33}C_{33 - r} = {}^{m}C_{r} \Rightarrow \ m = 33$.]$${\ ^{3}C_{3} \cdot \ ^{30}C_{30 - r} + \ ^{3}C_{2} \cdot \ ^{30}C_{31 - r} + \ ^{3}C_{1} \cdot \ ^{30}C_{32 - r} + \ ^{3}C_{0} \cdot \ ^{30}C_{33 - r} }{= \ ^{33}C_{33 - r} \equiv \ ^{m}C_{r} \equiv \ ^{m}C_{m - r} }{\Rightarrow m = 33 }$$

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