A small combination lock on a suitcase has 7 wheels, each labeled with the digits 0 to 9. How many 7 digit combinations are possible of no digit is repeated? If digits can be repeated? If successive digits must be different?

a) Use the multiplication principle to find the number of combinations with no repeated digits. First find N1, the number of possible outcomes for choosing the first digit. N1 equals 10 as there are 10 digits to choose from.

b) Use the multiplication principle to find the number combinations allowing repeated digits. First find N1, the number of possible outcomes for choosing the first digits. N1 equals 10 as there are 10 digits to choose from.

c) Use the multiplication principle to find the number combinations if successive digits must be different. First find N1, the number of possible outcomes for choosing the first digits. N1 equals 10 as there are 10 digits to choose from.

**Solution:-**

Multiplication Principle (for Counting)

In general, if n operations O1, O2,…On are performed in order, with possible number of outcomes N1,N2,…..Nn respectively, then there are

N1* N2*…..*Nn

Possible combine outcomes of the operations performed in the given order.

a) Use the multiplication principle to find the number of combinations with no repeated digits. First find N1, the number of possible outcomes for choosing the first digit. N1 equals 10 as there are 10 digits to choose from.

Find N2, the number of possible outcomes for choosing the second digit, given that the first digit cannot be repeated. N2 equals 9 as there is now one less digit to choose form, since the first cannot be repeated.

Find N3, the number of possible outcomes for choosing the third digit, given that the first and second digit cannot be repeated. N3 equals 8 as there are now two less digits to choose from, since the first two digits cannot be repeated.

Continue the above process up to N7 and use the multiplication principle to find the number of combinations with no repeated digits.

Number of combinations = 10*9*8*7*6*5*4 = 604800

b) Use the multiplication principle to find the number combinations allowing repeated digits. First find N1, the number of possible outcomes for choosing the first digits. N1 equals 10 as there are 10 digits to choose from.

Number of combinations = 10*10*10*10*10*10*10 = 10000000

c) Use the multiplication principle to find the number combinations if successive digits must be different. First find N1, the number of possible outcomes for choosing the first digits. N1 equals 10 as there are 10 digits to choose from.

Number of combinations = 10*9*9*9*9*9*9 = 5314410

Thanks, great help.