Counting, Probability & Sets› Probability, Sets & Worst-Case Selection
How many numbers are there between 100 and 300 which either begin with or end with 2 ?
Answer a.
The answer is (a) 110. There are 100 numbers beginning with 2 and 20 ending with 2, and 10 numbers do both, so the count is 100 + 20 minus 10 = 110. Every number between 100 and 300 has three digits, so beginning with 2 means the hundreds digit is 2, and ending with 2 means the units digit is 2. Count the first group: 200 to 299, which is 299 minus 200 plus 1 = 100 numbers. Count the second group: 102, 112, 122 and so on up to 292, rising in steps of 10, which is (292 minus 102) divided by 10, plus 1, that is 19 plus 1 = 20 numbers. The two groups overlap, because 202, 212 and so on up to 292 both begin and end with 2; that overlap is (292 minus 202) divided by 10, plus 1, which is 10 numbers. Simply adding 100 and 20 counts those 10 numbers twice, so subtract them once: 100 + 20 minus 10 = 110. The shortcut is to avoid the double count altogether by splitting into groups that cannot overlap: take all 100 numbers from 200 to 299, then add only the numbers ending in 2 that lie below 200, namely 102 up to 192, which is 10 numbers, giving 110 directly. That version needs one subtraction fewer and takes about twenty seconds, while listing the numbers out by hand would cost two or three minutes. The endpoints change nothing, since neither 100 nor 300 begins or ends with 2.
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