1. How many positive integers less than 1,000 do not have 7 as any digit?
(A) 700
(B) 728
(C) 736
(D) 770
(E) 819
2. Points A, B, and X do not all lie on the same line. Point X is 5 units from A and 3 units from B. How many other points in the sma eplane as A, B, and X are also 5 units from A and 3 units from B?
(A) None
(B) One
(C) Two
(D) Four
(E) More than four
1. Consider the numbers 000 - 999. We want to find the # of integers in this set that do not contain 7's. There are 7 choices for the first digit, 7 for the second, 7 for the third (two-digit numbers are represented by a leading 0). 7^3 = 729. However this includes 000, so we subtract 1, leaving 728, B.
2. Draw triangle ABX, with AX = 5, BX = 3. If you draw a circle of radius 5 centered at A, and a radius of circle 3 centered at B, you will see that the two circles intersect at X, as well as one other point (obtained by reflecting X across AB). Hence the answer is B. This assumes that A and B are already determined.
There's a likely typo in rspence's solution for (1)
The corrected version is:
1. Consider the numbers 000 - 999. We want to find the # of integers in this set that do not contain 7's. There are 9 choices (0,1,2,3,4,5,6,8,9) for the first digit, 9 for the second, 9 for the third (two-digit numbers are represented by a leading 0). 9^3 = 729. However this includes 000, so we subtract 1, leaving 728, B.
