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If n1a n dn2 are five-digit numbers, ...

If `n_1a n dn_2` are five-digit numbers, find the total number of ways of forming `n_1a n dn_2` so that these numbers can be added without carrying 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_(1) " and " n_(2)` can be added without carrying at any stage if `x_(i)+y_(i) le 9`.

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+..+8=36 ways as in this pair we cannot have 0 or 9. Thus, total number of ways is `36(55)^(4)`.
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