Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
The number of selections of 4 letters taken from the word "COLLEGE" must be:
18
22
Coefficient of $x^4$ in the expansion of $(1+x)^3(1+x+x^2)^2$
Coefficient of $x^4$ in the expansion of $(1+x)^2(1+x+x^2)^3$
Step-by-Step Solution
Key Concept: Convert the letter selection problem into a generating function where single letters contribute $(1+x)$ and repeated letters contribute $(1+x+x^2)$, then find the coefficient of $x^4$.
The word COLLEGE has letters: C(1), O(1), L(2), E(2), G(1). We need to select 4 letters considering repetitions. Cases: (i) All different: $\binom{5}{4}=5$ ways, (ii) One letter twice, two others: $\binom{2}{1}\binom{4}{2}=2\times 6=12$ ways (choose which of L or E appears twice, then pick 2 from remaining 4), (iii) Two pairs: $\binom{2}{2}=1$ way (both L and E twice). Total = $5+12+1=18$. This equals the coefficient of $x^4$ in $(1+x)^3(1+x+x^2)^2$ where $(1+x)^3$ represents the 3 single letters and $(1+x+x^2)^2$ represents the 2 repeated letters.
Correct Answer: 1,3