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n1a n dn2 are four-digit numbers, fin...

`n_1a n dn_2` are four-digit numbers, find the total number of ways of forming `n_1a n dn_2` so that `n_2` can be subtracted from `n_1` without borrowing at any stage.

Text Solution

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`n_(1)=x_(1)x_(2)x_(3)x_(4)x_(5)`
`n_(2)=y_(1)y_(2)y_(3)y_(4)y_(5)`
`n_(2)` can be subtracted from `n_(1)` without borrowing at any stage if `x_(i) ge y_(i)`.

Thus, `x_(5) " and " y_(5)` can be selected collectively by `10+9+8+..+1=55` ways. Similary, each pair `(x_(4),y_(4)),(x_(3),y_(3)),(x_(2),y_(2))` can be selected in 55 ways. But, pair `(x_(1),y_(1))` can be selected in 1+2+3+..+9=45 ways as in this pair we cannot have 0.
Thus, total number of ways is `45(55)^(4)`.
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