CBSE Class 11-science Answered

Is it with or without repetition of any of digits?

We'll assume without repetition.

Let the four digit number be, 1000w + 100y + 10y + z,

If we pick z = 1, then we can permute 3,5,7 and 9, taking three digits i.e. w, x and y in ⁴P₃ = 24 ways.

This means we will get 24 different numbers with unit digit as 1.

Similary we will get 24 different numbers with unit digit as 3, and so on.

Therefore, sum of all such numbers means,

24(1+3+5+7+9)

But this is the enumeration for unit digit only.

The same argument follows for other digits also, i.e. w, x and y.

Hence the sum of all the four digit numbers, formed without repetition of any digit, is,

24(1000+3000+5000+7000+9000)+24(100+300+500+700+900)+24(10+30+50+70+90)+24(1+3+5+7+9)

= 666600

The second example can be worked out similarly.

Regards,

Team,

TopperLearning.