1.

How many total numbers of seven-digit numbers can be formed having sum of whose digits is even is (a) 9000000 (b) 4500000 (c) 8100000 (d) 4400000 (e) None of these

Answer»

(b) Suppose x1 x2 x3 x4 x5 x6 x7 represents a seven digit
number. Then x1 takes the value 1, 2, 3, ....., 9 and x2, x3,
....., x7 all take values 0, 1, 2, 3, ......, 9.
If we keep x1, x2, ......, x6 fixed, then the sum x1 + x2 + ......
+ x6 is either even or odd. Since x7 takes 10 values 0, 1,
2, ....., 9, five of the numbers so formed will have sum of
digits even and 5 have sum odd.
Hence the required number of numbers
= 9 . 10 . 10 . 10 . 10 . 10 . 5 = 4500000.



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