Difference between revisions of "2007 AMC 12A Problems/Problem 16"

(Solution: add alternate solution with a little less listing out possibilities)
(Solution 2)
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Observe that, if the smallest and largest digit have the same parity, this uniquely determines the middle digit. If the smallest digit is not zero, then any choice of the smallest and largest digit gives <math>3! = 6</math> possible 3-digit numbers; otherwise, <math>4</math> possible 3-digit numbers. Hence we can do simple casework on whether 0 is in the number or not.
 
Observe that, if the smallest and largest digit have the same parity, this uniquely determines the middle digit. If the smallest digit is not zero, then any choice of the smallest and largest digit gives <math>3! = 6</math> possible 3-digit numbers; otherwise, <math>4</math> possible 3-digit numbers. Hence we can do simple casework on whether 0 is in the number or not.
  
[b]Case 1[/b]: 0 is not in the number. Then there are <math>\binom{5}{2} + \binom{4}{2} = 16</math> ways to choose two nonzero digits of the same parity, and each choice generates <math>3! = 6</math> 3-digit numbers, giving <math>16 \times 6 = 96</math> numbers.
+
Case 1: 0 is not in the number. Then there are <math>\binom{5}{2} + \binom{4}{2} = 16</math> ways to choose two nonzero digits of the same parity, and each choice generates <math>3! = 6</math> 3-digit numbers, giving <math>16 \times 6 = 96</math> numbers.
  
[b]Case 2[/b]: 0 is in the number. Then there are <math>4</math> ways to choose the largest digit (2, 4, 6, or 8), and each choice generates <math>4</math> 3-digit numbers, giving <math>4 \times 4 = 16</math> numbers.
+
Case 2: 0 is in the number. Then there are <math>4</math> ways to choose the largest digit (2, 4, 6, or 8), and each choice generates <math>4</math> 3-digit numbers, giving <math>4 \times 4 = 16</math> numbers.
  
 
Thus the total is <math>96 + 16 = 112 \Longrightarrow \mathrm{(C)}</math>. (by scrabbler94)
 
Thus the total is <math>96 + 16 = 112 \Longrightarrow \mathrm{(C)}</math>. (by scrabbler94)

Revision as of 11:53, 18 May 2018

Problems

How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two?

$\mathrm{(A)}\ 96\qquad \mathrm{(B)}\ 104\qquad \mathrm{(C)}\ 112\qquad \mathrm{(D)}\ 120\qquad \mathrm{(E)}\ 256$

Solution

We can find the number of increasing arithmetic sequences of length 3 possible from 0 to 9, and then find all the possible permutations of these sequences.

Common difference Sequences possible Number of sequences
1 $012, \ldots, 789$ 8
2 $024, \ldots, 579$ 6
3 $036, \ldots, 369$ 4
4 $048, \ldots, 159$ 2

This gives us a total of $2 + 4 + 6 + 8 = 20$ sequences. There are $3! = 6$ to permute these, for a total of $120$.

However, we note that the conditions of the problem require three-digit numbers, and hence our numbers cannot start with zero. There are $2! \cdot 4 = 8$ numbers which start with zero, so our answer is $120 - 8 = 112 \Longrightarrow \mathrm{(C)}$.

Solution 2

Observe that, if the smallest and largest digit have the same parity, this uniquely determines the middle digit. If the smallest digit is not zero, then any choice of the smallest and largest digit gives $3! = 6$ possible 3-digit numbers; otherwise, $4$ possible 3-digit numbers. Hence we can do simple casework on whether 0 is in the number or not.

Case 1: 0 is not in the number. Then there are $\binom{5}{2} + \binom{4}{2} = 16$ ways to choose two nonzero digits of the same parity, and each choice generates $3! = 6$ 3-digit numbers, giving $16 \times 6 = 96$ numbers.

Case 2: 0 is in the number. Then there are $4$ ways to choose the largest digit (2, 4, 6, or 8), and each choice generates $4$ 3-digit numbers, giving $4 \times 4 = 16$ numbers.

Thus the total is $96 + 16 = 112 \Longrightarrow \mathrm{(C)}$. (by scrabbler94)

See also

similar problem

2007 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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