Mathematics - Organized Counting and Permutations

1. On his university application, Enzo must list his course choices in order of preference. He must choose three of the four courses available in his major discipline, and two of the three courses offered in related subjects. In how many ways can Enzo list his course choices? Explain the reasoning for your answer.

2. How many ways are there to draw a 9 or a Jack from a standard deck of 52 playing cards.

3. a) Find the number of permutations of the letters in the word DIPLOMA.
b) How many ways are there of arranging the letters of DIPLOMA so that the letters O and I are together?

4. Show that the number of three-letter "words" that can be formed from the word CAMPGROUND is the same as the number of permutations of the letters of GROUND.

5. How many odd four-digit numbers, all of the digit different can be formed from the digits 0 to 7, if there must be a 4 in the number?

6. Solve the following equation for n:

7. A University has a telephone system in which extension numbers are three digits long with no repeated digits and no 0's. The university has 492 telephones at present and is planning to add another 35 in the near future.
a. Should the university change its system? Why or why not?
b. The Music Department uses extensions that begin with a 4. How many extensions can the Music Department have with the current system?

8. In how many ways can a 12-member soccer team share a half-time snack of 9 oranges and 3 apples if each member takes one piece of fruit? Explain your answer.