If the number of ways in which four distinct balls can be put into two identical boxes such that no box remains empty is equal to k, then k is4376

The problem is about combinations and permutations.

Step 1: We have 4 distinct balls and 2 identical boxes.

Step 2: We need to find the number of ways to distribute these balls into the boxes such that no box remains empty.

Step 3: Since the boxes are identical, we only care about the number of balls in each box, not which box they are in.

Step 4: The possible distributions are: 1 ball in one box and 3 balls in the other, or 2 balls in each box.

Step 5: For the first case, we choose 1 ball out of 4 to go in one box, and the remaining 3 automatically go in the other box. This can be done in 4C1 = 4 ways.

Step 6: For the second case, we choose 2 balls out of 4 to go in one box, and the remaining 2 automatically go in the other box. This can be done in 4C2 = 6 ways.

Step 7: So the total number of ways is 4 + 6 = 10.

Therefore, k = 10.

However, the statement says that k is equal to 4376, which is not correct based on the calculations above. There might be a mistake in the problem statement.