By Akiva Moiseevič Âglom; Isaak Moiseevič Âglom; James McCawley; Basil Gordon
ISBN-10: 0486655369
ISBN-13: 9780486655369
ISBN-10: 0486655377
ISBN-13: 9780486655376
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Extra info for Challenging mathematical problems with elementary solutions [Vol. I]
Sample text
This result of course coincides with that obtained in the first solution. It shows that among the numbers under consideration there are more with I's among their digits than without. = - 52 SOLUTIONS llb. Among the integers from I to 222,222,222 there are 22,222,222 ending in a 0 (namely, the numbers 10,20,30, ... , 222,222,220). In order to determine how many integers have a 0 in the next to last position, notice that what comes before this 0 can be anything from I to 2,222,222, while what comes after it can be anything from 0 to 9.
Then g is counted by each term of (I), and is therefore counted a net of I + I + I - I - I - I + I = I time. This analysis shows that expression (I) counts each element of A V B V C once. On the other hand, elements not in A V B V C are not counted in any of its terms, and therefore (I) is equal to #(A V B V C). 12c. The general case can be treated by the same reasoning as that used in part b. We must show that in the expression #(A 1) + #(AJ + ... + #(Am) - #(Al (\ AJ - #(Al (\ Aa) - ... - #(A mAm) + #(Al (\ A2 (\ Aa) + ...
Suppose that as N -- 00 the ratio q(N)IN approaches a limit; in this case this limit is called the probability that a number selected at random from the entire sequence has the desired property. Note that this probability depends on the way in which the numbers are arranged in a sequence. Changing the order of the numbers can change the value of the probability. Example: consider the positive integers arranged in increasing order: 1,2,3, ... Of the first N of these numbers, [N/2] are even; as N -- 00 the ratio [NI2]/N approaches i, which means that the probability that any number selected at random is even equals 1/2.
Challenging mathematical problems with elementary solutions [Vol. I] by Akiva Moiseevič Âglom; Isaak Moiseevič Âglom; James McCawley; Basil Gordon
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