JEE Questions & Answers

Step 1: Assign variables \( x_1, x_2, x_3, x_4 \) for the number of red pens each person gets, such that \( x_1 + x_2 + x_3 + x_4 = 10 \).

Step 2: The number of non-negative integer solutions for the equation can be found by the formula for combinations with repetition:

$$ \binom{n+k-1}{k-1} $$

Where \( n = 10 \) (total items) and \( k = 4 \) (number of groups).

$$ \binom{10+4-1}{4-1} = \binom{13}{3} = 286 $$

Step 3: Similarly, assign variables \( y_1, y_2, y_3, y_4 \) for the number of blue pens each person gets, such that \( y_1 + y_2 + y_3 + y_4 = 14 \).

Calculate the number of non-negative solutions for these equations:

$$ \binom{14+4-1}{4-1} = \binom{17}{3} = 680 $$

Step 4: Multiply the two results to find the total number of ways to distribute the pens:

$$ 286 \times 680 = 194,480 $$

Final Answer: 194,480