Permutations & Combinations
Permutations & Combinations
star_batch_jee_advanced_2025
Grade None
Question:
It $5$ letters are put in the $5$ envelopes. Find the no. of ways so that atleast $2$ letters are in wrong envelope:
Step-by-Step Solution
Key Concept: Converting a binomial series into an integral representation allows use of recurrence relations and the Wallis integral formula.
The sum $S = \sum_{r=0}^{n} \frac{(-1)^r C_r}{(4r+2)}$ is evaluated by recognizing that each term can be expressed as an integral: $\frac{(-1)^r C_r}{(4r+2)} = C_r \int_0^1 x(1-x^2)^r dx$. Summing the series under the integral gives $\int_0^1 x(1-x^2)^n dx = \frac{1}{2}\int_0^1 (1-x^2)^n dx$. Using the recurrence formula for $I_n = \int_0^1 (1-x^2)^n dx$, the result is $\frac{2^n n!}{(2n+1)!!}$.
Correct Answer: 2