A new design for an automotive door entry code lock consists

A new design for an automotive door entry code lock consists of the digits 1 through 6. How many unique entry codes are possible if

(a) Four numbers must be entered in sequence but any number can be repeated (e.g., 1-3-1-3).

(b) Four unique numbers must be entered two at a time (e.g., 1 and 3, 2 and 5).

Solution

the digits are from 1 to 6.

Number canbe repeated as normally the rottaion willbe for all digitis.

a) Four numbers must be in a sequence means

first number can be any of the 9 digits 1,2,.....9

II can be any of the 0,1,2.....

...

Hence total no of ways = 9x10³ =9000

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b) Four unique numbers 2 at a time.

Hence for first time we have to select first digit as non 0 and hence 9 ways and ii digit in the remaining 9 ways.

Hence 81 ways.

III and iv digits can be done in 10P2 ways = 90

Thus total no of ways = 81x90

= 7290