Difference between revisions of "1970 AHSME Problems/Problem 31"

Problem

If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 43, what is the probability that this number will be divisible by 11? $\text{(A) } \frac{2}{5}\quad \text{(B) } \frac{1}{5}\quad \text{(C) } \frac{1}{6}\quad \text{(D) } \frac{1}{11}\quad \text{(E) } \frac{1}{15}$

Solution

For the sums of the digits of be equal to 43, this means that the number's digits must fall under one of two categories. It either has 4 9's and a 7, or 3 9's and 2 8's. You can use casework from there to find that there are 15 5-digit numbers in which the sum of the digits is equal to 43. To find the number of ways that the number is divisible by 11, use the divisibility rule for 11, letting the number be represented by $abcde$. This gives 3 possible numbers that are divisible by 11, so the answer is $\frac{3}{15}=\fbox{B}$

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 