If `alpha=x_1,x_2,x_3 and beta=y_1,y_2,y_3` be two three digit numbers, then the number of pairs of `alpha and beta` that can be formed so that `alpha` can be subtracted from `beta` without borrowing.

Options

A. `2!xx10!xx10!`
B. `(45)(55)^(2)`
C. `3^(2)*5^(3)*11^(2)`
D. `136125

Answer

B, C, D

Explanation

Since, `alpha` can be subtracted from `beta` without borrowing, if `y_(i) ge x_(i),` for i=1,2,3.
let `x_(i)=lamda`
if i=1, then `lamda=1,2,3, . . .,9`
and if i=2 and 3, then `lamda=0,1,2,3, . . .,9`
Hence, total number of ways of choosing the pair `alpha,beta`
`=(underset(lamda=1)overset(9)(sum)(10-lamda))(underset(lamda=0)overset(9)(sum)(10-lamda))^(2)=(45)(55)^(2)`