Math 124 First Midterm Examination October1 1, 2010
NAME (Please print)
Page Points Score 2 20 3 20 4 20 5 20 6 20 Total 100
Instructions:
1. Do all computations on the examination paper. You may use the backs of the pages if necessary.
2. Put answers inside the boxes.
3. Please signify your adherence to the honor code:
I, , have neither given nor received aid in completion of this examination.
1 (20 Points) Score
1. (6 points) How many numbers between 1 and 10,000 have exactly one digit equal to 5?
Take any number from 0 to 999 not containing 5 ( 93 numbers ) and insert 5 in one of 4 positions. Alternatively, the number has either 1, 2, 3 or 4 digits. If one digit, there is one choice. If two digits, either the last digit is 5 and the first is neither 0 or 5, or the last digit is not 5 and the first is 5; so there are 8 + 9 = 17 choices. If three digits, either the last digit is 5 and the middle is not 5 and the first is not 0 or 5, or the middle digit is 5 and ...; there are 8×9+8×9+9×9 = 225 choices. If four digits, there are 8 × 9 × 9 + 8 × 9 × 9 + 8 × 9 × 9 + 9 × 9 × 9 = 2673 choices.
answer = 2916
2. (7 points) What is the number of ways to order the 26 letters of the alphabet so that no two of the vowels a, e, i, o and u occur consecutively? (Hint: order the consonants, then place the vowels.)
There are 21! ways to order the consonants. Then there are 22 places for the first vowel. Next, there are 23 − 2 = 21 places for the next vowel to appear somewhere either than next to the first vowel. Continue in this way.