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How many 3-digit numbers have the sum of their digits equal to 4?

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Bunuel
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How many 3-digit numbers have the sum of their digits equal to 4? [#permalink] New post   20 Jan 2020, 02:34
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64% (01:52) correct 36% (01:40) wrong based on 267 sessions
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How many 3-digit numbers have the sum of their digits equal to 4?

A. 4
B. 6
C. 7
D. 10
E. 12
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D
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CrackverbalGMAT
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Re: How many 3-digit numbers have the sum of their digits equal to 4? [#permalink] New post   20 Jan 2020, 05:23
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For a 3-digit number to have the sum of its digits equal to 4 (exactly), it can only begin with 1, 2, 3 or 4.

Case 1: Number begins with 1

1 _ _: remaining digits can be:

1a: 0 & 3 --> 2 such numbers
1b: 1 & 2 --> 2 such numbers

Therefore total number of such numbers possible = 4

Case 2: Number begins with 2

2 _ _: remaining digits can be:

2a: 0 & 2 --> 2 such numbers
2b: 1 & 1 --> 1 such number

Therefore total number of such numbers possible = 3

Case 3: Number begins with 3

3 _ _: remaining digits can only be 0 & 1 --> 2 such numbers possible

Case 4: Number begins with 4

4 _ _: remaining digits can only be 0 & 0 --> 1 such number possible

Therefore total number of such numbers possible = 4 + 3 + 2 + 1 = 10

Answer: (D)

Hope this helps.
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rajatchopra1994
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Re: How many 3-digit numbers have the sum of their digits equal to 4? [#permalink] New post   21 Jan 2020, 01:41
Number of cases:

103, 202, 301, 112, 121, 211, 130, 400, 310, 220.

Total Cases = 10

IMO-D
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InaKi20
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Re: How many 3-digit numbers have the sum of their digits equal to 4? [#permalink] New post   30 Jan 2020, 11:47
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Is there any faster method that can be applied?
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Re: How many 3-digit numbers have the sum of their digits equal to 4? [#permalink] New post   31 Jan 2020, 08:41
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Bunuel wrote:
How many 3-digit numbers have the sum of their digits equal to 4?

A. 4
B. 6
C. 7
D. 10
E. 12


We can separate into possible digits to select first then scramble the digits.
For example, 4 + 0 + 0 is one case, only 400 suffices being a 3-digit number.
3 + 1 + 0 -> 310, 301, 130, 103.
2 + 2 + 0 -> 220, 202.
2 + 1 + 1 -> 211, 121, 112.

Those are the only cases, thus there are 10 possible numbers. Notice how we break down from large digits into small digits one step at a time.

Ans: D
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Re: How many 3-digit numbers have the sum of their digits equal to 4? [#permalink] New post   06 Jan 2021, 06:00
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InaKi20 wrote:
Is there any faster method that can be applied?


Factorial might help.!

Digits summing to four = 211, 310 & 400.

211 can be rearranged in 3!/2! ways = 3.
310 can be rearranged in 3! ways = 6.
400 can be rearranged in one way only as replacing 4 from 1st position will lead to the failure of the condition i.e 'It should be a three digit integer'.

Total Ways :- 6+3+1 = 10.
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Re: How many 3-digit numbers have the sum of their digits equal to 4? [#permalink] New post   20 Feb 2024, 06:37
InaKi20 wrote:
Is there any faster method that can be applied?

­yes ! Watch this video for the quiekest way to solve this type of problem (1) Permutations and Combinations 10 (Similar to Different Distribution) - YouTube
 
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MihirBathia
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How many 3-digit numbers have the sum of their digits equal to 4? [#permalink] New post   03 Mar 2024, 04:34
Looking at the options I got the answer, but in such problems, I am always on the verge of getting on my horse and calculating the number of terms in the AP, where a(first term)=103, d(difference)=9 and tn (last term)=994. The problem is, I understand 994 also to be a number with the sum 4=9+9+4, 22=2+2=4. Basically I end up calculating the seed of the number.­ Please help
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Re: How many 3-digit numbers have the sum of their digits equal to 4? [#permalink] New post   03 Mar 2024, 04:45
ElninoEffect wrote:
InaKi20 wrote:
Is there any faster method that can be applied?


Factorial might help.!

Digits summing to four = 211, 310 & 400.

211 can be rearranged in 3!/2! ways = 3.
310 can be rearranged in 3! ways = 6.
400 can be rearranged in one way only as replacing 4 from 1st position will lead to the failure of the condition i.e 'It should be a three digit integer'.

Total Ways :- 6+3+1 = 10.



It seems you made an error when saying 310 can be rearranged in 3! Ways, as the leftmost digit can’t be 0. Also, you missed one more case where digits could be 220.

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Re: How many 3-digit numbers have the sum of their digits equal to 4? [#permalink]
03 Mar 2024, 04:45
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