# Permutation/Combination

Find all the numbers from 1 to 999 in which all the digits are different.

The registration number of a car consists of 3 letters of the alphabet followed by an interger between 1 and 999 inclusive followed by a letter of the alphabet, e.g. ABC6D, TUF27M, ACD123Y. Given that all 26 letters of the alphabet may be used and that any letter or digit may be repeated, find the total number of different registration numbers that could be formed.

Find also in how many ways of these registration numbers the letters and digits are all different.

ans: 738,999x26^4 , 23x24x25x26x738

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1)Find all the numbers from 1 to 999 in which all the digits are different.

For 3 digits numbers : The leading digit can be 1 to 9 (9 cases)

The 2nd digit can be 0 to 9 but no repeat the leading digit (9 cases)

The last digit can be 0 to 9 but no repeat other digits (8 cases)

9 * 9 * 8 = 648 numbers

For 2 digits numbers : 9*9 = 81 numbers

For 1 digit numbers : 1 - 9 total 9 numbers

There are 648 + 81 + 9 = 738 numbers

2)The leading 3 letters have 26^3 cases

An integer have 999 cases

The last letter have 26 cases

Total (26^3) (999) (26) = 999 x 26^4 different registration numbers that could be formed.

3)Find also in how many ways of these registration numbers the letters and digits are all different.

The leading 3 letters have 26 x 25 x 24 cases

An integer have 738 cases (By question 1 result.)

The last letter have 23 cases ( no repeat with first 3 letters)

Total (26 x 25 x 24) x (738) x (23) = 23x24x25x26x738 registration numbers the letters and digits are all different.