# Worksheet: Counting Outcomes with Restrictions

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

Q1:

James’s password must be five characters long. He can use the digits 0 to 9, and cannot use the same digit more than once. How many different passwords could James create?

Q2:

In how many ways can an odd number of 6 digits be formed using the numbers if no digits are to be repeated?

Q3:

How many three-digit even numbers, with no repeated digits, can be formed using the elements of the sets ?

Q4:

In how many ways can a 4-digit even number, with no repeated digits, be formed using the elements of the set ?

Q5:

How many four-digit numbers, with no repeated digits, can be formed using the elements of the set ?

Q6:

How many three-digit numbers, which have an even tens digit and no repeated digits, can be formed using the elements of the set ?

Q7:

In how many ways can a three-digit number, starting with an even digit and containing no repeated digits, be formed from the numbers ?

Q8:

How many two-digit numbers, which end with the digit 2 and have no repeated digits, can be formed using the elements of the set ?

Q9:

In how many ways can a 3-digit number, which is greater than 300 and has no repeated digits, be formed using the numbers ?

Q10:

How many three-digit numbers, which are less than 900 and have no repeated digits, can be formed using the elements of the set ?

Q11:

How many three-digit numbers can be formed by picking the units digit from the set , the tens digit from the set , and the hundreds digit from the set ?

Q12:

In how many ways can a 5-digit code be formed using the numbers 1 to 9? Note, the code can have repeated digits.

Q13:

Phone numbers on a particular network are twelve-digit long, where the first three digits are always 072. Calculate the total number of different phone numbers which the network can use.

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, Mohammed 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.

Q17:

Find the number of ways to form a 2-digit number, with no repeated digits, given 4 diﬀerent digits to choice from.

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?