Final answer:
To find the number of strings of length 8 with exactly 2 a's and 2 b's, we can use the concept of combinations. After calculating the combinations, we need to subtract the cases where the two a's or two b's are in the same position. The final answer is 406.
Step-by-step explanation:
To determine the number of strings of length 8 that have exactly 2 a's and 2 b's, we can use the concept of combinations. We have 8 positions to fill, and we need to choose 2 positions for the a's and 2 positions for the b's. This can be calculated using the formula C(n, k) = n! / (k!(n-k)!), where n is the total number of positions and k is the number of positions to choose. Plugging in the values, we get C(8, 2) * C(6, 2) = 28 * 15 = 420. However, this overcounts the cases where the two a's or two b's are in the same position. To correct for this, we subtract the number of strings where both a's are in the same position and the number of strings where both b's are in the same position. There are 8 possibilities for the positions of the two a's and 6 possibilities for the positions of the two b's, so the final answer is 420 - 8 - 6 = 406.