Question

A manufacturer assigns a ten-character serial number to every product it produces. Assume that the first two characters and the last character in the serial number are letters from the English alphabet, and the remaining characters are allowed to be any of the numbers 0 - 9.

a.)How many possible serial numbers can be produced assuming repetitions are allowed?

b.): How many possible serial numbers can be produced, allowing repetitions, if the first letter is a vowel (not including Y), the last letter is selected from{W,X,Y,Z}, the fifth character is the number 2, the ninth character is the number 0, and the remaining numbers are odd?

c.)Re-do part b but assume that repeats are NOT allowed.

*remember that sometimes Y plays as a vowel.

Answer 1

a) Number of characters in the serial number = 3

Number of numbers in the serial number = 7

No. of available choices for characters = 26

No. of available choices for numbers = 10

Total number of possible serial numbers

b) No. choices for the first letter = 5

No. choices for the last letter = 4

The fifth character is fixed = 2

9th character is fixed = 0

The remaining numbers are odd.

Number of choices for odd numbers = 5

Hence, after these conditions, the total possible serial numbers are (each number in the given product expression below corresponds to the number of choices for each position in the serial number. For fixed positions, there is only 1 choice)

c) After repetition is not allowed

The number of choices for the first and the last letter will remain the same since they are disjoint sets and have different set elements.

The number of choices for the second character will be 24 since two alphabets are taken for the first and the last character and repetition is not allowed.

The 5 and 9th numbers are 2 and 0.

The rest of the positions are 3rd, 4th, 6th,7th, and 8th. For these 5 positions, we have 5 odd numbers 1, 3, 5, 7, and 9. Since repetition is not allowed hence these 5 odd numbers will be permuted among themselves within these 5 positions. Total number of ways in which this can be done = 5! = 120

Hence, after repetition is not allowed, the total number of possible serial numbers are

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