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VP  P
Joined: 07 Dec 2014
Posts: 1206
How many three digit numbers are there whose first digit equals the su  [#permalink]

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9 00:00

Difficulty:   75% (hard)

Question Stats: 51% (02:31) correct 49% (02:26) wrong based on 98 sessions

### HideShow timer Statistics How many three digit numbers are there whose first digit equals the sum of the second and third digits?

A. 18
B. 27
C. 36
D. 45
E. 54
Math Expert V
Joined: 02 Aug 2009
Posts: 7764
Re: How many three digit numbers are there whose first digit equals the su  [#permalink]

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gracie wrote:
How many three digit numbers are there whose first digit equals the sum of the second and third digits?

A. 18
B. 27
C. 36
D. 45
E. 54

In such questions, it is good to write down the ways as it does not entail too much of work...
Let the number be xyz
1) x is 9, so y+z=9
Ways : (9,0); (8,1);(7,2);(6,3);(5,4) ..
Each of the above 5 ways can be taken as 2 different arrangements.. 981 or 918
So 5*2=10
2) x is 8, so y+z=8
Ways : (8,0);(7,1);(6,2);(5,3);(4,4) ..
4 of the above 5 ways can be taken as 2 different arrangements..
So 4*2+1=9
3) x is 7, so y+z=7
Ways : (7,0);(6,1);(5,2);(4,3)
Each of the above 4 ways can be taken as 2 different arrangements..
So 4*2
4) x is 6, so y+z=6
Ways : (6,0);(5,1);(4,2);(3,3)....
3 of the above 4 ways can be taken as 2 different arrangements..
So 3*2+1=7
And so on
There is a pattern...
So 10+9+8+7+6+5+4+3+2=54
Last 2 is when x is 1...
101 or 110

E
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Senior SC Moderator V
Joined: 22 May 2016
Posts: 3066
How many three digit numbers are there whose first digit equals the su  [#permalink]

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gracie wrote:
How many three digit numbers are there whose first digit equals the sum of the second and third digits?

A. 18
B. 27
C. 36
D. 45
E. 54

After you write the first digit, write all the second digits in ascending order, then all the third digits in descending order.

101
110

202
211
220

303
312
321
330

404
413
422
431
440

The pattern is
100s have 2 such numbers
200s have 3 " "
300s have 4
400s have 5

900s will have 10

Total = sum of integers 2 - 10 inclusive
(9 groups, and each group has +1 more than its first digit))

SUM of consecutive integers?
(Average) * (# of terms, n)
Average = $$\frac{First+ Last}{2}$$

$$n=$$(Last-First) + 1 =
[(10 -2 ) + 1]= 9

($$\frac{First+ Last}{2}*n)$$

SUM: $$\frac{2+10}{2}*9)=(6*9)=54$$

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Intern  B
Joined: 01 Dec 2018
Posts: 11
Re: How many three digit numbers are there whose first digit equals the su  [#permalink]

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generis wrote:
gracie wrote:
How many three digit numbers are there whose first digit equals the sum of the second and third digits?

A. 18
B. 27
C. 36
D. 45
E. 54

After you write the first digit, write all the second digits in ascending order, then all the third digits in descending order.

101
110

202
211
220

303
312
321
330

404
413
422
431
440

The pattern is
100s have 2 such numbers
200s have 3 " "
300s have 4
400s have 5

900s will have 10

Total = sum of integers 2 - 10 inclusive
(9 groups, and each group has +1 more than its first digit))

SUM of consecutive integers?
(Average) * (# of terms, n)
Average = $$\frac{First+ Last}{2}$$

$$n=$$(Last-First) + 1 =
[(10 -2 ) + 1]= 9

($$\frac{First+ Last}{2}*n)$$

SUM: $$\frac{2+10}{2}*9)=(6*9)=54$$ after the highlighted portion we can simply use Sn formula for athematic progression. Answer in 1 step. Re: How many three digit numbers are there whose first digit equals the su   [#permalink] 14 Dec 2018, 22:54
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