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06 Dec 2008, 19:32
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This question is well beyond the GMAT (I'd say it's a 900-level question 
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
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
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06 Dec 2008, 20:14
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my answer is 56, although i am not happy about how i came to it..
#of 1 digit numbers: 1
For 2 digits we have (2,3),(4,1)
#of 2 digit numbers: 4 + 1 (this one is derived from adding a 0 to the 1 digit number) = 5
For 3 digits we have (1,1,3)(1,2,2)
#of 3 digit numbers: 3!/2!+3!/2!+ 1 (500) + 4*2 (adding one zero to the 2 digit numbers - 2 ways, in between or at the end) = 15
For 4 digits we have (1,1,1,2)
#of 4 digit numbers: 4!/3! + 1 (5000) + 4*3 (adding 2 zeroes to the 2 digit numbers - 3 ways, in between, at the end, or one in between and one at the end) + 6*3 (adding one 0 to the 3 digit numbers - 3 ways, after 1st digit, after 2nd digit, after 3rd digit) = 35
total = 56
took me 4+ mins - i do hope its not a GMAT Q!!
#of 1 digit numbers: 1
For 2 digits we have (2,3),(4,1)
#of 2 digit numbers: 4 + 1 (this one is derived from adding a 0 to the 1 digit number) = 5
For 3 digits we have (1,1,3)(1,2,2)
#of 3 digit numbers: 3!/2!+3!/2!+ 1 (500) + 4*2 (adding one zero to the 2 digit numbers - 2 ways, in between or at the end) = 15
For 4 digits we have (1,1,1,2)
#of 4 digit numbers: 4!/3! + 1 (5000) + 4*3 (adding 2 zeroes to the 2 digit numbers - 3 ways, in between, at the end, or one in between and one at the end) + 6*3 (adding one 0 to the 3 digit numbers - 3 ways, after 1st digit, after 2nd digit, after 3rd digit) = 35
total = 56
took me 4+ mins - i do hope its not a GMAT Q!!
06 Dec 2008, 20:50
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I choose D,it also took me 8 mins to solve
positive integers less than 10,000 => the number has 4 digits
made of (5,0,0,0), (1,1,1,2), (4,1,0,0), (3,2,0,0),(3,1,1,0),(2,2,1,0)
(5,0,0,0) => 5000, 0500, 0050, 0005 => 4
Similar to (1,1,1,2) =>4
(4,1,0,0): as 1,4 can be interchangable
4100
4010
4001
0410
0401
0041
=> 6 * 2 = 12
Similar to (3,2,0,0) => 12
Similar to (3,1,1,0) => 12
Similar to (2,2,1,0) => 12
(As they have the same pattern (x,y,z,z))
Total = 4 + 4 + 12 + 12 + 12 + 12 = 56
positive integers less than 10,000 => the number has 4 digits
made of (5,0,0,0), (1,1,1,2), (4,1,0,0), (3,2,0,0),(3,1,1,0),(2,2,1,0)
(5,0,0,0) => 5000, 0500, 0050, 0005 => 4
Similar to (1,1,1,2) =>4
(4,1,0,0): as 1,4 can be interchangable
4100
4010
4001
0410
0401
0041
=> 6 * 2 = 12
Similar to (3,2,0,0) => 12
Similar to (3,1,1,0) => 12
Similar to (2,2,1,0) => 12
(As they have the same pattern (x,y,z,z))
Total = 4 + 4 + 12 + 12 + 12 + 12 = 56
