A44 If a positive integer n is factored into primes in the form p\' ···, then th

Question

A44 If a positive integer n is factored into primes in the form p' ···, then the divisors of n have the form p p.., where each exponent bi can range from 0 to a. For example, the divisors of 12 (which is 2231) have the form 2n3b, where bi = 0, 1 or 2, and b2-0 or 1 . This fact allows us to count the number of divisors of a given integer. (a) List the six divisors of 12, including 1 and 12, and explain combinatorially (that is, by a counting procedure established in this section) why there are six of them. (b) Without actually listing them, find the number of divisors of each of the integers 77, 100, 800, and 3000. A45 (a) If a three-letter word that uses letters from {A, B, C is selected at random, find the probability that the first letter is not A, the second is not B, and the third is not C. (b) If a rearrangement of ABC is selected at random, find the probability that the first letter is not A, the second is not B, and the third is not C.

Explanation / Answer

A45 . b)we have three letters which are A B and C .ie

We can arrange these letters in 3P3=6 Since it is regarding the arrangement we can use permutation ways S={ABC ,CAB,BCA,BAC,CBA,ACB}

here p(A is not the first letter)=2/6=1/3

p( B is not first)=2/6=1/3

p(c is not first)=1/3

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