How many positive 3-digit integers have the sum of their digits less

Question

How many positive 3-digit integers have the sum of their digits less than 5?

A. 17
B. 18
C. 19
D. 20
E. 21

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nick1816 · Answer

Suppose 3-digit number = 'abc'
where a is positive integer and b and c are non-negative integer.

a+b+c < 5

a+b+c+k=5, where k is positive integer

A+1+b+c+K+1 = 5, where A and K are non-negative integer

A+b+c+K=3

Total possible solution = 6C3 = 20

GMATWhizTeam · Answer

Solution 
 Three digits are same 
 • Only one case: 111 
  Two digits are same 
 • Case 1: integers has anyone of (1, 2, 3, or 4) and two 0s.
 o Number of such integers  = 4C1*1*1 = 4  
• Case 2: Integers consists of two similar digits and one 0 
 o Number of such integers  = 2C1*1*2*1 = 2*2 = 4  
• Case 3: Integers consists of two same and one different digit(1, 1, 2)
 o Number of such integers  = \frac{3!}{2! }= 3   
 All three digits are different 
 • Case 1: Integer consists of (1, 3,0)
 o Number of such integers  = 2*2*1 = 4  
• Case 2: Integer consists of (1, 2,0)
 o Number of such integers  = 2*2*1 = 4   
So total number of possible integers  = 1+ 4+4+3+4+4 = 20 
Thus, the correct answer is  Option D.

chetan2u · Answer

The sum of digits <5, so only 3 possibilities

SUM=4

1) Digits are 4, 0, 0
First place can be taken by 1, and remaining two by 0.
Only one number can be made 400

2) Digits are 3, 1, 0.....say number is XYZ
Hundreds digit, X, can be taken by 3 or 1....Tens by remaining one and 0 => 2*2=4

3) Digits are 2, 1, 1.....say number is XYZ
Ways to arrange these =3!/2!=3 ways

4) Digits are 2, 2, 0.....say number is XYZ
Ways to arrange these =2 ways

SUM=3

1) Digits are 3, 0, 0
First place can be taken by 1, and remaining two by 0.
Only one number can be made 300

2) Digits are 2, 1, 0.....say number is XYZ
Hundreds digit, X, can be taken by 2 or 1....Tens by remaining one and 0 => 2*2=4

3) Digits are 1, 1, 1.....Only 1 way

SUM=2

1) Digits are 2, 0, 0
First place can be taken by 1, and remaining two by 0.
Only one number can be made 200

2) Digits are 1, 1, 0.....say number is XYZ
Hundreds digit, X, can be taken by 1....Tens by remaining one and 0 => 1*2=2

SUM=1

1) Digits are 1, 0, 0
First place can be taken by 1, and remaining two by 0.
Only one number can be made 100

Total 1+4+3+1+4+1+1+2+1+2= 20.

Or 
Keep it very simple
you can just check for hundreds, tens, and ones digit as given in attached figure

sambitspm · Answer

Sum 1
100 - 1 possibility

Sum 2
101 -2 possibility
200 -1 possibility

sum 3
111 - 1 possibility
120 - 4 possibility
300 - 1 possibility

Sum 4
112 - 3 possibility
130 -4 possibility
220 - 2 possibility
400 - 1 possibility

I missed the 300 possibility and ended up with 19.  Answer is E - 20

poojakhanduja3017 · Answer

Interesting question for beginners. The answer should be D:20
 
CONCEPT: Basically, the question asks how many 3-digit numbers (excluding those in the form 0XX, and 00X )have digits which add up to 4. We can take the question like: we know that there is a total of 5 to be divided among the 3 digits, we just have to determine the number of ways it can be divided. 
 
Asuuming the number is  abc 
Step 1: Out of these, a can't be 0. 
Step 2: a+b+c+x=5, where k is a positive integer
Step 3: And now x can't be 0. 
Step 4: Standardizing the equation: A+1+b+c+X+1 = 5, where A and X are non-negative integer
Step 5: A+b+c+K=3
Step 6 : Now, each of these can be 0. Picking up any 3 numbers out of the six will be  6C3  = 20

Aalto700 · Answer

400 _ 1 possibility
310 _ 2*2*1 possibility 310, 301, 103,101 
211 _ 3*2*1/2! possibility 211, 121, 112
220 _ 2*2*1/2! possibilities 220, 202

300 - 1
210 - 2*2*1
111 - 1

200 - 1
110 - 2
 
100 - 1

1 + 4 + 3 + 2
+ 1+4+1
+ 1+2
+1
 = 20   a boring question!

Aalto700 · Answer

Nice idea, but from 5 to 6 does not seem correct. What's wrong  Krunaal  ?

zaltinator · Answer

I understand the solution till A+b+c+K=3, but then why is 6C3 done? Is there a standard formula that has been applied, and if not, how?

How many positive 3-digit integers have the sum of their digits less

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GMAT Problem Solving (PS) Questions

How many positive 3-digit integers have the sum of their digits less

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