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How many positive integers less than 10,000 are there in which the sum of the digits equals 5?

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How many positive integers less than 10,000 are there in which the sum of the digits equals 5?
(A) 31
(B) 51
(C) 56
(D) 62
(E) 93
Official Answer

1 Best Answer

Best answer
i think the "stars and bars" method is probably what they're getting at. this will be a 1, 2, 3, or 4 digit number, so we'll consider the cases separately:
1 digit
can only be 5
2 digits
can be 14, 23, 32, 41, or 50. we can get this result by considering a string of five stars that represent adding up to 5, and by putting a separator (a bar) somewhere to distinguish the first digit from the second. Thus our 2-digit numbers can look like this
*|**** = 14
**|*** = 23
***|** = 32
****|* = 41
*****| = 50
Note that we can't start with a bar because that would mean the first digit is 0 (note that ending with a bar, as in the case of 50, means the final digit is 0). Once we set the first position to be a star, we have 4 stars and 1 bar to put in any order in the remaining 5 positions. This gives us 5!/(4!1!)=5, which corresponds to what we generated manually.
For three digit numbers, we have 5 stars and 2 bars, some examples of which include
**|*|** = 212
*||**** = 104
Again, the first position must be a star, but the remaining 6 will consist of 4 stars and 2 bars, giving us 6!/(4!2!)=15.
For four digit numbers, we have 5 stars and 3 bars. First position is again a star leaving 7 positions to arrange 4 stars and 3 bars, giving us 7!/(4!3!)=35.
Adding these up we get 1+5+15+35=56.
answered Dec 5, 2014 by Student (41 points)
selected Feb 15, 2015 by
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Jyotsna Mehta

Indian School of Business, Co'18

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