Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
For the equation $x + y + z + w = 19$, the number of positive integral solutions is equal to:
The number of ways in which 15 identical things can be distributed among 4 persons
The number of ways in which 19 identical things can be distributed among 4 persons
Coefficient of $x^{19}$ in $(x^0 + x^1 + ... + x^{19})^4$
Coefficient of $x^{19}$ in $(x + x^2 + x^3 + ... + x^{19})^4$
Step-by-Step Solution
Key Concept: Positive integral solutions require a shift to non-negative integers, reducing the sum from 19 to 15, which connects to distributing 15 items among 4 recipients.
We need positive integral solutions to $x + y + z + w = 19$ where $x, y, z, w \geq 1$. Using the substitution $x' = x-1, y' = y-1, z' = z-1, w' = w-1$, we get $x' + y' + z' + w' = 15$ where $x', y', z', w' \geq 0$. By stars and bars, this equals $\binom{15+4-1}{4-1} = \binom{18}{3}$, which is the number of ways to distribute 15 identical things among 4 persons (Option 1). Alternatively, coefficient of $x^{19}$ in $(x + x^2 + ... + x^{19})^4 = x^4(1 + x + ... + x^{18})^4$ gives coefficient of $x^{15}$ in $(1 + x + ... + x^{18})^4$, which equals Option 4 directly (Option 4).
Correct Answer: 1,4