If (2n+1)Pn-12nPn = 1121, then n^2 + n + 15 is equal to: — Solution | JEE Mathematics

NVAMediumJEE 2023Permutations (P(n,r))

If $\frac{(2n+1)P_{n-1}}{2nP_n} = \frac{11}{21}$ , then $n^2 + n + 15$ is equal to:

Answer

Correct answer:45

Step-by-step solution

Standard Method

Given: $\frac{(2n+1)P_{n-1}}{2nP_n} = \frac{11}{21}$

Find: $n^2 + n + 15$

Using the permutation form shown in the extracted working,

\frac{(n+2)!(2n-1)!}{(2n+1)!(n-1)!} = \frac{21}{11}

This simplifies to

\frac{(n+2)(n+1)n}{(2n+1)(2n)} = \frac{21}{11}

Cancelling the common factor $n$ and simplifying further,

\frac{(n+1)(n+2)}{2n+1} = \frac{42}{11}

From the solution, this gives

n = 5

Now substitute $n = 5$ into the required expression:

n^2 + n + 15 = 25 + 5 + 15 = 45

Therefore, the required numerical value is $45$ .

Detailed Solution Working

Given: $\frac{(2n+1)P_{n-1}}{2nP_n} = \frac{11}{21}$

Find: $n^2 + n + 15$

The solution concludes that the correct answer is $45$ . Using the displayed factorial simplification in the page:

\frac{(n+2)!(2n-1)!}{(2n+1)!(n-1)!} = \frac{21}{11}

Expanding factorial terms,

\frac{(n+2)(n+1)n}{(2n+1)(2n)} = \frac{21}{11}

\frac{(n+1)(n+2)}{2n+1} = \frac{42}{11}

From the provided working,

n = 5

Hence,

n^2 + n + 15 = 5^2 + 5 + 15 = 25 + 5 + 15 = 45

So, the correct answer is $45$ .

Direct Substitution After Simplification

Given: $\frac{(2n+1)P_{n-1}}{2nP_n} = \frac{11}{21}$

Find: $n^2 + n + 15$

The quickest route is to convert the permutation ratio into a factorial expression and cancel common terms immediately. The extracted solution reduces the ratio to an algebraic form that yields

n = 5

Once $n$ is known, substitute directly:

n^2 + n + 15 = 5^2 + 5 + 15 = 45

Therefore, the required value is $45$ .

Common mistakes

Using the permutation formula incorrectly. A common error is to substitute $nP_r$ without keeping track of both the upper value and the selected number. Always write $nP_r = \frac{n!}{(n-r)!}$ carefully before simplifying.
Cancelling factorial terms too early or in the wrong direction. This can change the ratio incorrectly. Expand only the needed factors, then cancel common terms systematically.
Finding $n$ correctly but substituting wrongly into $n^2 + n + 15$ . After obtaining $n = 5$ , evaluate each term carefully as $25 + 5 + 15$ , not $5^2 + 15$ or any partial substitution.

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