Question

A computer randomly chooses a binary string of length 6. What is the probability that exactly two characters in the string turn out to be 1s?

Answer 1

Let us use the formula for number of permutation with repetitions to find all strings of length 6 that exactly two characters in the strings turn out to be 1s. It follows that there are $P_6(2,4)=\frac{6!}{2!4!}=\frac{6\cdot 5}{2}=15$ such strings. Taking into account that the total number of strings of length 6 is equal to $2^6=64,$ we conclude that the probability $p$ that exactly two characters in the string turn out to be 1s is $p=\frac{15}{64}.$