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Scrambling

OK.  I must redeem myself in the realm of counting!  Too many hasty errors on my last post for discrete math and not enough “thinking.”  Don’t fall victim to this.  And now to the question:

Q:  How many bit strings (reminder: a bit is either 0 or 1) of length 10 have:

(a) Exactly 3 zeros?
(b) Have the same number of 0 and 1
(c) Have at least 7 zeros
(d) Have at least 3 ones?

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How many…

Q:  How many positive integers less than 1000 are:
(a) divisible by 7?
(b) divisible by 7 but not by 11?
(c) divisible by either 7 or 11?
(d) divisible by exactly one of 7 or 11?
(e) divisible neither by 7 or 11?
(f) have distinct digits?
(g) Have distinct digits and are even?

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How many ways

Qs:

(a) A multiple-choice test contains 10 questions. There are four possible answers to each question. How many ways can a student answer the questions on the test if the student answers every question?

(b) In the scenario of (a):  How many ways can a student answer the questions on the test if the student can leave answers blank?

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