k = 0 6 C 3 51 - k is equal to [2023]

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Q.
$\sum_{k = 0}^{6} {}^{51 - k}{C_{3}}$ is equal to [2023]

1 ${}^{51}{C_{3} -} {}^{45}{C_{3}}$

2 ${}^{52}{C_{4} -} {}^{45}{C_{4}}$

3 ${}^{52}{C_{3} -} {}^{45}{C_{3}}$

4 ${}^{51}{C_{4} -} {}^{45}{C_{4}}$

Ans.
(2)

$\sum_{k = 0}^{6} {}^{51 - k}C_{3}$ $= {}^{51}C_{3} + {}^{50}C_{3} + {}^{49}C_{3} + {}^{48}C_{3} + {}^{47}C_{3} + {}^{46}C_{3} + {}^{45}C_{3}$

$As we have, {}^{n}C_{r} + {}^{n}C_{r - 1} = {}^{n + 1}C_{r} \Rightarrow {}^{n}C_{r - 1} = {}^{n + 1}C_{r} - {}^{n}C_{r}$

$So, {}^{51}C_{3} = {}^{52}C_{4} - {}^{51}C_{4}, {}^{50}C_{3} = {}^{51}C_{4} - {}^{50}C_{4}$ and ${}^{45}C_{3} = {}^{46}C_{4} - {}^{45}C_{4}$

$\Rightarrow \sum_{k = 0}^{6} {}^{51 - k}C_{3}$ $= {}^{52}C_{4} - {}^{51}C_{4} + {}^{51}C_{4} - {}^{50}C_{4} + \dots + {}^{47}C_{4} - {}^{46}C_{4} + {}^{46}C_{4} - {}^{45}C_{4}$

$= {}^{52}C_{4} - {}^{45}C_{4}$

Solution

Q.
$\sum_{k = 0}^{6} {}^{51 - k}{C_{3}}$ is equal to [2023]

1 ${}^{51}{C_{3} -} {}^{45}{C_{3}}$

2 ${}^{52}{C_{4} -} {}^{45}{C_{4}}$

3 ${}^{52}{C_{3} -} {}^{45}{C_{3}}$

4 ${}^{51}{C_{4} -} {}^{45}{C_{4}}$

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Solution

Q. ∑k=06C351-k is equal to [2023]

1 C3-51C345

2 C4-52C445

3 C3-52C345

4 C4-51C445

Ans. (2) ∑k=06C351-k =C351+C350+C349+C348+C347+C346+C345 As we have, Crn+Cr-1n=Crn+1⇒Cr-1n=Crn+1-Crn So, C351=C452-C451,C350=C451-C450 and C345=C446-C445 ⇒∑k=06C351-k=C452-C451+C451-C450+…+C447-C446+C446-C445 =C452-C445

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Q.
$\sum_{k = 0}^{6} {}^{51 - k}{C_{3}}$ is equal to [2023]

1 ${}^{51}{C_{3} -} {}^{45}{C_{3}}$

2 ${}^{52}{C_{4} -} {}^{45}{C_{4}}$

3 ${}^{52}{C_{3} -} {}^{45}{C_{3}}$

4 ${}^{51}{C_{4} -} {}^{45}{C_{4}}$