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: Set up the distribution condition.
Total pens: \[ 10+14=24 \] Since there are: \[ 4 \] persons and each gets: \[ 6 \] pens, \[ 4\times6=24 \] Let the number of red pens received by the four persons be: \[ x_1,x_2,x_3,x_4 \] Then: \[ x_1+x_2+x_3+x_4=10 \] Also each person gets exactly: \[ 6 \] pens, so blue pens are automatically determined. Thus: \[ 0\le x_i\le6 \]

Step 2: Count non-negative integer solutions.
We need the number of solutions of: \[ x_1+x_2+x_3+x_4=10 \] with: \[ x_i\le6 \] Total non-negative solutions: \[ \binom{10+4-1}{4-1} = \binom{13}{3} \] \[ =286 \]

Step 3: Subtract invalid cases.
Invalid cases occur when some: \[ x_i\ge7 \] Suppose: \[ x_1\ge7 \] Put: \[ x_1'=x_1-7 \] Then: \[ x_1'+x_2+x_3+x_4=3 \] Number of solutions: \[ \binom{3+4-1}{3} = \binom63 = 20 \] Similarly for each variable. Thus total invalid cases: \[ 4\times20=80 \] No overlap possible because: \[ 7+7>10 \]

Step 4: Find total valid distributions.
\[ 286-80 \] \[ =206 \]

Step 5: Identify the final answer.
Therefore: \[ \boxed{206} \]