# Exercises - Counting

1. If you have 2 pairs of shoes, 3 pants, and 5 t-shirts, how many different outfits can you make?

Fundamental Counting Principle.

There are $2 \times 3 \times 5 = 30$ such outfits.

2. How many ways can you answer a 5-question True/False test?

Fundamental Counting Principle.

There are two choices for each question, so $2 \times 2 \times 2 \times 2 \times 2 = 32$ ways are possible.

3. How many 5-digit numbers are divisible by 5? (Note: the presence of "leading zeros" does not count towards the number of digits. So for example, 01234 is a 4-digit number. Also, you may use a digit more than once.)

Fundamental Counting Principle.

There are $9$ choices for the first digit (which can't be zero), $2$ choices for the last digit (which must be $0$ or $5$) and $10$ choices for each of the remaining digits (since they can be anything). So, there are $9 \times 10 \times 10 \times 10 \times 2$ possible such numbers.

4. How many 4-digit odd numbers are there, if repeated digits are allowed? What if there are no repeated digits?

Fundamental Counting Principle.

Repeated digits allowed: There are $9$ possibilities for the first digit (since it can't be zero), $10$ possibilities for the second and third digits (since they can be anything), and $5$ possibilities for the last digit (since it must be odd). Thus there are $9 \times 10 \times 10 \times 5 = 4500$ such numbers.

Repeated digits not allowed: Pick the last digit first, then the first, then the other two to limit the number of cases that must be considered. There are $5$ choices for the last digit (since it must be odd), $8$ choices for the first digit (since it must not be zero or the last [non-zero] digit), $8$ choices for the second digit (since it must not equal the first or last digit), and $7$ choices for the third digit (since it must not equal the first, second, or last digits). Thus, there are $8 \times 8 \times 7 \times 5 = 2240$ such numbers.

5. How many ways can you arrange 4 different pictures in a row on a wall?

Fundamental Counting Principle. $4 \times 3 \times 2 \times 1 = 24$.

6. How many ways can you arrange 4 chemistry books, 3 math books, and 2 physics books on a shelf? What if you want to keep books of the same subject together?

Assumming the books of any one subject are indistinguishable, there are $$\frac{9!}{4! \cdot 3! \cdot 2!} = 1260 \textrm{ ways}$$

7. How many ways can you arrange 4 pictures in a row on a wall if you have 7 pictures to choose from?

Use Fundamental Counting Principle or permutations: $7 \times 6 \times 5 \times 4 = {}_7 P_4 = 840$ ways
8. How many ways can we award 1st, 2nd, and 3rd place in an 8-person race?

Fundamental Counting Principle. $8 \times 7 \times 6 = 336$ ways

9. How many ways can you arrange the letters in the word MISSISSIPPI?

Permutations of indistinguishable objects. There are $4$ I's, $4$ S's, $2$ P's, in $11$ letters, so there are $$\frac{11!}{4! \cdot 4! \cdot 2!} = 34650 \textrm{ ways}$$

10. How many ways can you arrange the letters of the word STATISTICS?

Permutations of indistinguishable objects. There are $3$ S's, $3$ T's, $2$ I's, in $10$ letters, so there are $$\frac{10!}{3! \cdot 3! \cdot 2!} \textrm{ ways}$$

11. How many ways can you arrange 4 pennies and 4 nickels in a row?

Permutations of indistinguishable objects. There are $4$ pennies, $4$ nickels, in a row of $8$ coins, so there are $$\frac{8!}{4! \cdot 4!} = 70\textrm{ ways}$$

12. A club has 20 members. How many ways can we choose a 4-person committee?

Combinations. There are ${}_{20} C_4 = 4845$ ways.

13. A club has 20 members. How many ways can we choose a president, a vice-president, a secretary, and 4 senators?

Choose the president ($20$ choices), then the vice-president ($19$ choices), then the secretary ($18$ choices), then the $4$ senators (${}_{17} C_4$ choices). So there are $$20 \cdot 19 \cdot 18 \cdot {}_{17} C_4 = 16279200 \textrm{ ways}$$

14. A club has 12 freshmen and 8 sophomores.

1. How many ways can we choose a 4-person committee of 2 freshmen and 2 sophomores?
2. How many ways can we choose a 4-person committee with at least 2 freshmen?

15. A jar contains 8 red, 9 blue, and 10 white balls

1. In how many ways can someone pick 3 balls and get one of each color?
2. In how many ways can someone pick 5 balls and get 3 red and 2 blue?
3. In how many ways can someone pick 5 balls and get exactly 3 red balls?
4. In how many ways can someone pick 2 balls, both of the same color?