A U.S. Social Security Number (SSN) is a nine-digit number. The first digit(s) a

Question

A U.S. Social Security Number (SSN) is a nine-digit number. The first digit(s) are allowed to be 0. (a) How many SSNs are available? (b) How many of these are even? (c) How many have all of their digits even? (d) How many read the same backward and forward? (e.g. 122989221) (e) How many have none of their digits equal to 8? (f) How many have exactly one digit equal to 8? In how many ways can a black rook and a white rook be placed on different squares on a chess board such that neither is attacking the other? (In other words, on a standard, 8 times 8 chess board, in how many ways can you pick two different squares the first square for the black rook and the second square for the white rook-on the board such that they are on different rows and different columns?)

Explanation / Answer

SSN number : 9 digits from 0,1,2,4,5,6,7,8,9

a. How many SSNs are available

we have 9 place to fill and each postion can have 10 choices (number from 0,1,2,3,4,5,6,7,8,9)

So the total available SSNs = 10x10x10 x10x10x10 x10x10x10 =109

b. How many of them are even

For number to be even the last digit (units place) to be one of (0,2,4,6,8)

we have 9 place to fill and 1 to 8 places can have 10 choices (number from 0,1,2,3,4,5,6,7,8,9) and last place has 5 choices

So the total SSNs which are even = 10x10x10 x10x10x10 x10x10x 5 =108x5

C.How many have all their digits as even :

For SSNs to have all their digit as even : we have 9 place to fill and each postion can have 5 choices (number from 0,2,4,6,8)

So the total SSNs having all their digits are even = 5x5x5x 5x5x5 x5x5x5 = 59

D.How many read the same backward and forward :

To have the SSNs read the same backward and forward : We can have choices for the first 5 digits as the last four digits to be the same as first 4 digits:

So, each of the five digits have 10 choices(0,1,2,3,4,5,6,7,8,9)

So the Total number of SSNs that can read the same backward and forward = 10x10x10x10x10 = 105

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