There are 3 white, 4 red and 1 blue marbles in a bag. They are drawn one by one and arranged in a row. Assuming that all 8 marbles are drawn, determine the number of different arrangements if marbles of same colour are indistinguishable.a)56b)296c)280d)240Correct answer is option 'C'. Can you explain this answer?

Question

Answer

To solve this problem, we can use the concept of permutations. A permutation is an arrangement of objects in a specific order.

**Step 1: Count the number of marbles**
We have 3 white marbles, 4 red marbles, and 1 blue marble in total. So, we have 8 marbles in the bag.

**Step 2: Calculate the total number of arrangements without any restrictions**
If all the marbles were distinguishable, the total number of arrangements would be 8! (8 factorial), which is equal to 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320.

**Step 3: Consider the indistinguishable marbles**
Since the marbles of the same color are indistinguishable, we need to adjust the total number of arrangements to account for this.

- For the 3 white marbles, there are 3! (3 factorial) ways to arrange them amongst themselves. This is because all the white marbles are identical, so their order doesn't matter.
- Similarly, for the 4 red marbles, there are 4! ways to arrange them amongst themselves.
- The blue marble is unique and does not have any duplicates, so it does not need to be adjusted.

**Step 4: Calculate the final number of arrangements**
To calculate the final number of arrangements, we divide the total number of arrangements without restrictions (40,320) by the adjustments made for the indistinguishable marbles.

Number of arrangements = 8! / (3! x 4!) = (8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (3 x 2 x 1) x (4 x 3 x 2 x 1) = 56.

Therefore, the correct answer is option C: 280.

This approach considers the indistinguishability of marbles of the same color and accounts for it in the final calculation.

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