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## #1 2013-12-07 20:21:54

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

### eight-digits numbers...

Find the number of 8-digit numbers the sum of whose digits is 4.

I am confused as I got the result as 120 and some of my friends told me they got 149.
Is their any formula to find it.

pls help.

Niharika

friendship is tan 90°.

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## #2 2013-12-07 21:08:48

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: eight-digits numbers...

Hi niharika_kumar;

You are correct, 120 is the answer.

The answer is done using generating functions but can also be done playing spot the pattern.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #3 2013-12-07 22:05:56

Nehushtan
Member
Registered: 2013-03-09
Posts: 894
Website

### Re: eight-digits numbers...

The possible combinations of nonzero digits in such an 8-digit number are as follows:

(i) {4}
(ii) {3,1}
(iii) {2,2}
(iv) {2,1,1}
(v) {1,1,1,1}

We take each case in turn.

(i) The only possible 8-digit number is 40000000.

(ii) The leading digit must be 1 or 3, and the other digit can be placed in any of the other 7 places. Thus there are 7 + 7 = 14 such 8-digit numbers.

(iii) One 2 is the leading digit and the other 2 can be placed in any of the other 7 places, so number of such 8-digit numbers is 7.

(iv) If the leading digit is 2, the two 1s can be placed in the other places in [sup]7[/sup]C[sub]2[/sub] = 21 ways. If the leading digit is 1, the other two digits can be placed in the other places in [sup]7[/sup]P[sub]2[/sub] = 42 ways. ∴ Number of such 8-digit numbers = 21 + 42 = 63.

(v) One of the 1s is the leading digit and the other 3 can be placed in the other places in [sup]7[/sup]C[sub]3[/sub] = 35 ways.

Hence the total number of such 8-digit numbers is 1 + 14 + 7 + 63 + 35 = 120.

Last edited by Nehushtan (2013-12-07 22:06:35)

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## #4 2013-12-08 23:01:20

niharika_kumar
Member
From: Numeraland
Registered: 2013-02-12
Posts: 1,062

### Re: eight-digits numbers...

thank you so much.
I really forgot that we could solve it using combination and permutation.

friendship is tan 90°.

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