What is Q3 mean? Someone can help me solve Q3? Thanks. A password is a string of

Question

What is Q3 mean? Someone can help me solve Q3? Thanks.

A password is a string of ten characters, where each character is a lowercase letter, a digit, or one of the eight special characters !, @, #, $, %, &, and (, and). A password is called awesome, if it contains at least one digit or at least one special character. Determine the number of awesome passwords. Determine the number of functions f: {1, 2, 3, 4) rightarrow {a, b, c, ..., z}, such that f(1) = f(2), or f(3) = f(4), or f(1) notequalto f(3). Let n greaterthanorequalto 1 be an integer. Assume that n is odd. Determine the number of bitstrings of length n that contain more 0's than 1's. Justify your answer in plain English and at most three sentences. Assume that n is even. Determine the number of bitstrings of length n in which the number of 0's is equal to the number of 1's. Determine the number of bitstrings of length n that contain strictly more 0's than 1's.

Explanation / Answer

Solution

f : {1, 2, 3, 4} ---à {a, b, ….., z} should mean f(1), f(2), f(3), f(4) = one of the letters of the alphabet.

Now, f(1) can be any one of the 26 letters and so there are 26 possibilities for f(1) …… (1)

Once f(1) is fixed, f(2) cannot take any different alphabet since we are given f(1) = f(2) and hence there is only 1 possibility for f(2)………………………………………………… (2)

Given f(1) f(3), once f(1) is fixed, f(3) can take only anyone of the remaining 25 letters… (3)

Again, once f(3) is fixed, f(4) cannot take any different alphabet since we are given f(3) = f(4) and hence there is only 1 possibility for f(4)………………………………………………… (4)

Thus, from (1), (2), (3), (4), f : {1, 2, 3, 4} ---à {a, b, ….., z} can take 26x1x25x1 = 650 possible values . or there are 650 such functions. ANSWER

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