Supposed a 4digit PIN must be formed using the digits 0 thro

Supposed a 4-digit PIN must be formed using the digits 0 through 9.

1) How many such PINS do not repeat a digit consecutively? ie. 9129 does not repeat a digit consecutively,

even though it repeats the digit 9. However, 3341 does repeat the digit 3 consecutively.

2) How many such PINS have exactly two 2s?

3) How man such PINS have at least two 2s?

4) 5885 is a palindrome, but 1231 is not a palindrome. List every such PIN that is a palindrome. How many are there? Can you think of a more clever way to count them without list every single one?

Solution

(1)

total number of such PINS = 10*9*9*9 = 7290

(2)

= (4c2)(9*9)

= 486

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(3)

At least 2 2\'s = (exactly two 2\'s) + (excatly 3 2\'s) + (exactly four 2\'s)

= ( 486 ) + ( 4c3*9 ) + 1

= 486 + 36 + 1

= 523

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(4)

= 10*9*1*1

= 90