Q:

how many five digit palindromes are there where the digit β€œ1” appears at least once?

Accepted Solution

A:
A 5 digit palindrome number must be written as[tex] xyzyx [/tex]where [tex] x, y, z [/tex] are digits. We want "1" to appear at least once, which leads to the following cases:Only one of the variable is equal to 1 (the other two are not)In this case, the possible patterns of the numbers are [tex] 1yzy1,\ x1z1x,\ xy1yx [/tex]. We have 9 choices for each of the non-one digits (all the reamaining digits). So, the count of these kind of number is[tex] (9\cdot 9)\cdot 3 = 243[/tex]Two variables are equal to 1 (the third is not)In this case, the possible patterns of the numbers are [tex] 11z11,\ 1y1y1,\ x111x [/tex]. We have 9 choices for the non-one digit (all the reamaining digits). So, the count of these kind of number is[tex] 9\cdot 3 = 27[/tex]All variables are equal to 1In this trivial case we only have one number, [tex] 11111 [/tex]Total: [tex] 243+27+1 = 271 [/tex]