k=0^6 ^(51-k)C3: is equal to — Solution | JEE Mathematics

MCQEasyJEE 2023Applications of P&C

$\sum_{k=0}^{6} {}^{(51-k)}C_3$ : is equal to

A
${}^{51}C_4 - {}^{45}C_4$
B
${}^{51}C_3 - {}^{45}C_3$
C
${}^{52}C_4 - {}^{45}C_4$
D
${}^{52}C_3 - {}^{45}C_3$

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given:

\sum_{k=0}^{6} \binom{51-k}{3}

Find: The equivalent combination expression and the correct option.

Rewrite the summation as

\binom{51}{3}+\binom{50}{3}+\binom{49}{3}+\cdots+\binom{45}{3}

Use the identity

\sum_{r=a}^{b} \binom{r}{p}=\binom{b+1}{p+1}-\binom{a}{p+1}

with $a=45$ , $b=51$ and $p=3$ .

Therefore,

\sum_{k=0}^{6} \binom{51-k}{3}=\binom{52}{4}-\binom{45}{4}

Detailed Working and Source Discrepancy

Given:

\sum_{k=0}^{6} \binom{51-k}{3}

Find: The correct option.

Expanding in reverse order gives

\binom{45}{3}+\binom{46}{3}+\cdots+\binom{51}{3}

Now apply the hockey-stick identity repeatedly:

\binom{n}{r}+\binom{n}{r-1}=\binom{n+1}{r}

which leads to the finite sum formula

\sum_{r=45}^{51} \binom{r}{3}=\binom{52}{4}-\binom{45}{4}

Hence the expression matches option C in the provided options. The solution text states "The Correct Option is B", but its own working and final boxed answer are

\binom{52}{4}-\binom{45}{4}

which corresponds to option C. Therefore the mathematically correct answer is C.

Common mistakes

Using the wrong summation identity. Students may write $\sum \binom{r}{3}=\binom{51}{4}-\binom{45}{4}$ by not increasing the upper index by $1$ . In the hockey-stick identity, the upper limit contributes $b+1$ , so use $\binom{52}{4}$ .
Matching the result to the wrong option label. The source solution text says option B, but the worked result is $\binom{52}{4}-\binom{45}{4}$ , which is actually option C. Always compare the final expression with the listed options.
Expanding the sum in the wrong range. Since $k=0$ to $6$ , the terms are $\binom{51}{3},\binom{50}{3},\ldots,\binom{45}{3}$ . Missing the last term or starting from the wrong value changes the result.

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Answer

Step-by-step solution

Common mistakes

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