# Worksheet: Counting Outcomes with Restrictions

In this worksheet, we will practice finding the number of outcomes in a probability problem with a condition.

**Q14: **

In a gallery, each painting is referenced by two distinct English letters, and a four-digit number which has no zeros and no repeated digits. How many paintings can be referenced using this system?

**Q15: **

Determine the number of ways a 7-digit number can be formed from 7 different non-zero digits, given that each digit can NOT be used more than once.

**Q16: **

After a recent reorganisation, Jacob is taking over responsibility for the manufacturing of odd numbers on the house sign number production line. As part of his scientific investigation into production levels, he wants to know how many three digit numbers only contain odd digits. Calculate the answer for him.

**Q18: **

Without repeating any of the digits, determine how many numbers can be formed from the digits of the number 54,321, and find out how many of those numbers start with the digit 4 and end with the digit 1.

- A5, 3
- B120, 60
- C25, 9
- D120, 9
- E120, 6

**Q19: **

A building has 5 doors which are numbered as . Determine the number of ways a person can enter, and then leave the building, if they cannot use the same door twice.

**Q20: **

Without repeating any of the digits, determine how many four-digit numbers can be formed from , and find out how many of those numbers start with the digit 6.

- A720, 24
- B360, 60
- C720, 60
- D360, 120

**Q21: **

A new school is going to assign a 7-digit ID number to each of its students, where each digit is a number from 0 to 9 and no digit can be repeated. Find the probability that a randomly generated ID number is 7,654,321.

- A
- B
- C
- D
- E

**Q22: **

Eleven people, nine men and two women, arrive at the theater at the same time and have to line up to buy tickets. What is the probability that a woman would be at each end of the line if they lined up in random order?

- A
- B
- C
- D
- E

**Q23: **

In how many ways can 3 prizes be awarded to 9 people if a person can win more than one prize?

**Q24: **

In how many ways can 9 people sit in 9 seats arranged in a circle?

**Q25: **

Let . How many numbers can be formed from the elements of if no digit is used more than once and no number has more than 3 digits?