Question

In how many ways can 5 identical red balls and 3 identical blue balls be arranged in a row?

• A. 56
• B. 70
• C. 126
• D. 210

Hint:

Use multinomial coefficient for identical objects.

Answer 1

The Correct Option is B

To find the number of ways to arrange 5 identical red balls and 3 identical blue balls in a row, we can use the formula for permutations of multiset. The formula for arranging a total of \( n \) items where there are groups of identical items is given by:

\[ \frac{n!}{n_1! \times n_2! \times \cdots \times n_k!} \]

where \( n_1, n_2, \ldots, n_k \) are the sizes of the identical groups.

1. In this problem, the total number of balls \( n \) is 8 (5 red balls + 3 blue balls).
2. The number of identical red balls \( n_1 \) is 5.
3. The number of identical blue balls \( n_2 \) is 3.
4. Substituting into the formula, we get:

\[ \frac{8!}{5! \times 3!} = \frac{40320}{120 \times 6} = \frac{40320}{720} = 56 \]

Thus, the number of ways to arrange 5 identical red balls and 3 identical blue balls in a row is 56.