Question

How many bit strings of length 10 contain at least four 1s?

A) 252
B) 330
C) 420
D) 512

Answer 1

To find the number of bit strings of length 10 that contain at least four 1s, we subtract the number of bit strings without four 1s from the total number of possible bit strings. The number of bit strings without four 1s can be found by subtracting the combinations with three 1s, two 2s, and one 3 from the total combinations.

Step-by-step explanation:

The easiest way to approach this problem is to find the number of bit strings that do not contain at least four 1s, and then subtract that from the total number of possible bit strings of length 10.

To find the number of bit strings without four 1s, we subtract the combinations with exactly three 1s, two 2s, and three 3s from the total number of combinations.

So, the number of bit strings of length 10 that contain at least four 1s is given by:

1. The total number of bit strings of length 10, which is 2^10 = 1024
2. Subtract the number of bit strings with exactly three 1s, which is 10 choose 3 = 120
3. Subtract the number of bit strings with exactly two 2s, which is 10 choose 2 = 45
4. Subtract the number of bit strings with exactly one 3, which is 10 choose 1 = 10

Therefore, the number of bit strings of length 10 that contain at least four 1s is 1024 - 120 - 45 - 10 = 849.