Question

A licence plate consists of seven symbols: digits or letters. How many licence plates are there if the following is true:

(1) there must be three letters and four digits, and symbols may repeat?

(2) no restrictions on the quantity of letters and numbers, and symbols may repeat?

Answer 1

Please note

Also, by the rule of permutations, the number of arrangements of n things, all taken together (repetition allowed) = nⁿ.

There are 26 (A - Z) letters and 10 single unit digits (0 - 9)

There is no restriction on the place of the symbols and repetition is allowed.

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(1) Choose 3 letters from 26 in 26C3 = 2600 ways

Choose 4 digits in 10C4 = 210 ways

These 7 chosen symbols can be arranged in 7⁷ ways.

Therefore Total Number of license plates = 2600 * 210 * 7⁷ = 4.49654 * 10¹¹.

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(2) No restrictions on quantity. Therefore total possible options = 26 letters + 10 digits = 36

Therefore each of the 7 positions can hold any of these 36 symbol.

Therefore total number of arrangements = 36⁷ = 78,364,164,096

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