Answer :
These 7 bits has 2 options, the total number of bit strings is 2^7 = 128. Using the combination formula, the total number of bit strings is C(7,1) = 7. and the total number of bit strings is C(7,3) = 35.
To find the number of bit strings of length 10 that begin with 101, we need to consider that the first three bits are fixed (101). Therefore, we have 7 remaining bits to fill with either 0 or 1. Since each of these 7 bits has 2 options, the total number of bit strings is 2^7 = 128.
To find the number of bit strings of length 10 that have a weight of 3 and begin with 101, we first note that we already have two 1s in the string (101). So, we need to place 1 more 1 in the remaining 7 bits. We can choose any one of these 7 positions for this 1. Using the combination formula, the total number of bit strings is C(7,1) = 7.
To find the number of bit strings of length 10 that have a weight of 5 and begin with 101, we first note that we already have two 1s in the string (101). We need to place 3 more 1s in the remaining 7 bits. We can choose any three of these 7 positions for the 1s. Using the combination formula, the total number of bit strings is C(7,3) = 35.
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