The number of ways to distribute 30 identical candies among four children C1, C2, C3 and C4 so that

Q: Why can't we just use the stars and bars formula directly?

The stars and bars formula $\binom{n+k-1}{k-1}$ is a powerful tool, but it only works for lower-bound constraints (where variables are $\ge 0$). When you have upper-bound constraints, the simple formula overcounts. You must use the inclusion-exclusion principle or generating functions to subtract the cases that violate those upper limits.

Q: Why did the total number of candies change from 30 to 24?

We pre-distributed the minimum required candies to $C_2$ and $C_3$. By giving 4 to $C_2$ and 2 to $C_3$, we effectively 'used up' 6 candies. Since the total was 30, we are left with $30 - 6 = 24$ candies to distribute freely among the four children under the new, shifted constraints.

Q: Is there a faster way than generating functions?

You could use the inclusion-exclusion principle directly on the stars and bars formula. The generating function method is essentially a systematic way to perform inclusion-exclusion, ensuring you don't miss any overlapping cases. For JEE Advanced, mastering generating functions is safer and more robust for complex constraints.

Question

The number of ways to distribute 30 identical candies among four children $C_1, C_2, C_3$ and $C_4$ so that $C_2$ receives atleast 4 and atmost 7 candies, $C_3$ receives atleast 2 and atmost 6 candies, is equal to

Options:

(A) 205

(B) 615

(C) 510

(D) 430

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The problem asks us to distribute 30 identical candies among four children $C_1, C_2, C_3,$ and $C_4$. We represent this with the equation:
$$x_1 + x_2 + x_3 + x_4 = 30$$

The constraints are given as $4 \le x_2 \le 7$ and $2 \le x_3 \le 6$, while $x_1, x_4 \ge 0$. To simplify, we handle the lower bounds by pre-allocating the candies.

Let $y_2 = x_2 - 4$ and $y_3 = x_3 - 2$. Substituting these into the original equation:
$$x_1 + (y_2 + 4) + (y_3 + 2) + x_4 = 30$$

This simplifies to the reduced equation:
$$x_1 + y_2 + y_3 + x_4 = 24$$

The new constraints are $0 \le y_2 \le 3$ and $0 \le y_3 \le 4$, with $x_1, x_4 \ge 0$.

### The Power of Generating Functions

We represent the distribution possibilities for each child as a polynomial. For children with no upper limit ($C_1$ and $C_4$), the generating function is:
$$f(x) = 1 + x + x^2 + \dots = (1-x)^{-1}$$

For $C_2$, the constraint $0 \le y_2 \le 3$ yields:
$$g(x) = 1 + x + x^2 + x^3 = \frac{1-x^4}{1-x}$$

For $C_3$, the constraint $0 \le y_3 \le 4$ yields:
$$h(x) = 1 + x + x^2 + x^3 + x^4 = \frac{1-x^5}{1-x}$$

The total generating function $P(x)$ is the product of these individual functions:
$$P(x) = (1-x)^{-1} \cdot \frac{1-x^4}{1-x} \cdot \frac{1-x^5}{1-x} \cdot (1-x)^{-1}$$

Simplifying this expression, we obtain:
$$P(x) = (1-x^4)(1-x^5)(1-x)^{-4}$$

### The Final Extraction

We seek the coefficient of $x^{24}$ in the expansion of $P(x)$. First, we expand the numerator:
$$(1-x^4)(1-x^5) = 1 - x^4 - x^5 + x^9$$

We multiply this by the series expansion of $(1-x)^{-4}$. Using the generalized binomial theorem, the coefficient of $x^r$ in $(1-x)^{-n}$ is $\binom{n+r-1}{n-1}$. With $n=4$, the coefficient is $\binom{r+3}{3}$.

We now extract the coefficient of $x^{24}$ from the product $(1 - x^4 - x^5 + x^9) \sum_{r=0}^{\infty} \binom{r+3}{3} x^r$:
$$	ext{Coeff} = \binom{24+3}{3} - \binom{20+3}{3} - \binom{19+3}{3} + \binom{15+3}{3}$$

This simplifies to:
$$\binom{27}{3} - \binom{23}{3} - \binom{22}{3} + \binom{18}{3}$$

Calculating the individual binomial coefficients:
$$\binom{27}{3} = \frac{27 	imes 26 	imes 25}{3 	imes 2 	imes 1} = 2925$$
$$\binom{23}{3} = \frac{23 	imes 22 	imes 21}{3 	imes 2 	imes 1} = 1771$$
$$\binom{22}{3} = \frac{22 	imes 21 	imes 20}{3 	imes 2 	imes 1} = 1540$$
$$\binom{18}{3} = \frac{18 	imes 17 	imes 16}{3 	imes 2 	imes 1} = 816$$

Performing the final arithmetic:
$$2925 - 1771 - 1540 + 816 = 430$$

The total number of ways to distribute the candies is **430**.

Animated Solution for Mathematics - Permutations and Combinations: The number of ways to distribute 30 identical candies among four children $C_{1}, C_{2}, C_{3}$ and $C_{4}$ so that $C_{2}$ receives atleast 4 and atmost 7 candies, $C_{3}$ receives atleast 2 and atmost 6 candies, is equal to

Select Answer:

Visualized Solution

Problem Visualization

Formulating the Equation

Variable Substitution

Simplified Equation

New Constraints

Generating Function Setup

Simplifying the Expression

Polynomial Expansion

Coefficient Formula

Applying the Formula

Writing the Combinations

Calculating Combinations

Final Result

The Sigma Insight: Combinations and Selection

Analyzing the Setup

The Power of Generating Functions

The Final Extraction

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