Question

The number of ways to distribute \(10\) identical red pens and \(14\) identical blue pens among four persons such that each person gets \(6\) pens is ________.

Hint:

Number of non-negative integer solutions of: \[ x_1+x_2+\cdots+x_n=r \] is: \[ \binom{r+n-1}{n-1} \]

Answer 1

Correct Answer: 206

Step 1: Understanding the Concept:
We have a total of \( 10 + 14 = 24 \) pens to be distributed among 4 people. Since each person must receive exactly 6 pens, and the total number of pens is 24, the total pens each person receives is fixed. The distribution is determined solely by the number of red pens each person gets, as the number of blue pens will then be automatically fixed (\( b_i = 6 - r_i \)). Since the pens of the same color are identical, this is a partition problem with constraints.
Step 2: Key Formula or Approach:
Let \( r_i \) be the number of red pens given to the \( i \)-th person.
We must satisfy:
1) \( r_1 + r_2 + r_3 + r_4 = 10 \)
2) \( 0 \le r_i \le 6 \) (since a person can receive at most 6 pens in total)
The number of ways is the coefficient of \( x^{10} \) in the expansion of \( (1 + x + x^2 + x^3 + x^4 + x^5 + x^6)^4 \).
Step 3: Detailed Explanation:
The generating function is:
\[ P(x) = (1+x+x^2+x^3+x^4+x^5+x^6)^4 = \left(\frac{1-x^7}{1-x}\right)^4 = (1-x^7)^4 (1-x)^{-4} \]
Expanding the terms using Binomial Theorem:
\[ (1-x^7)^4 = 1 - 4x^7 + 6x^{14} - \dots \]
\[ (1-x)^{-4} = \sum_{n=0}^{\infty} \binom{n+4-1}{4-1} x^n = \sum_{n=0}^{\infty} \binom{n+3}{3} x^n \]
We need the coefficient of \( x^{10} \):
\[ \text{Coeff of } x^{10} = 1 \cdot (\text{Coeff of } x^{10} \text{ in } (1-x)^{-4}) - 4 \cdot (\text{Coeff of } x^3 \text{ in } (1-x)^{-4}) \]
\[ = \binom{10+3}{3} - 4 \binom{3+3}{3} \]
\[ = \binom{13}{3} - 4 \binom{6}{3} \]
Calculating the values:
\[ \binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 13 \times 2 \times 11 = 286 \]
\[ \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]
\[ \text{Total ways} = 286 - 4(20) = 286 - 80 = 206 \]
Step 4: Final Answer:
The number of ways is 206.