Find the number of `n` digit numbers, which contain the digits 2 and 7, but not the digits 0, 1, 8, 9.

Explanation

The total number without any restrictions containing digits 2,3,4,5,6,7 is `n(U)=6^(n)`
The total number of numbers that contains 3,4,5,6,7 is
`n(A)=5^(n)`
the total number of numbers that contain 2,3,4,5,6 is
`n(B)=5^(n)`. ,brgt the total number of numbers that contain 3,4,5,6 is
`n(AcapB)=4^(n)`.
The total number of numbers that do not contain digit 2 and 7 is `n(A cup)`
i.e., `n(AcupB)=n(A)+n(B)-n(AcapB)`
`=5^(n)+5^(n)4^(n)=2(5^(n))-4^(n)`
Hence, the total number of numbers that contain 2 and 7 is
`n(A'cupB')`
`thereforen(A'capB')=n(U)-n(AcupB)=6^(n)-2*(5^(n))+4^(n)`