Question

Shari has an electronic entry system on her house. To enter, one must input a six character code where the first three entries are letters of the alphabet (A through Z) and the last three are single digit numbers (0 through 9). If repeated letters/numbers are not allowed, how many different codes are possible?

Answer 1

• The first 3 entries of the 6 character code are letters of the alphabet (A through Z) and the last 3 entries are numbers (0 through 9)
• Repetition is not allowed

To find the first entry:

Since the first entry is an alphabet, n = 26 (total number of letters)

Now in order to select one among 26 letters we use

Hence the number of ways in which the first entry can be selected is 26 ways.

To find the second entry:

We know that the second entry is also an alphabet.

But now we cannot use all 26 letters of the alphabet since repetition is not allowed and we have used one letter of the alphabet for the first entry.

Hence, n = 25

Now in order to find 1 among 25 letters we use

Hence the number of ways in which the second entry can be selected is 25 ways.

To find the third entry:

Now since 2 letters of the alphabet have been used, we have only 24 letters left.

In order to find 1 among 24 letters we use

Hence the number of ways in which the third entry can be selected is 24 ways.

To find the fourth entry:

Same concept applies for numbers.

We have, n = 10 (total number of numbers from 0 through 9)

To select one among 10 we use

Hence the number of ways in which the fourth entry can be selected is 10 ways.

To find the fifth entry:

We now only have n = 9 total numbers left as one has been used as the fourth entry.

To select one among 9 we use

Hence the number of ways in which the fifth entry can be selected is 9 ways.

To find the sixth entry:

We now have only n = 8 numbers left as two of the ten numbers have been used as the fourth and the fifth entry.

To select one among 8 we use

Hence the number of ways in which the sixth entry can be selected is 8 ways.

To find the number of possible codes:

Total number of possible codes = Total number of possible codes = 26 x 25 x 24 x 10 x 9 x 8 = 11,232,000

Hence, there are 11,232,000 possible codes with no repetition allowed.

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