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Difference between revisions of "1998 AIME Problems/Problem 7"

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Another way we can approach this problem is by imagining a line of 98 stones. We want to place the stones into <math>4</math> boxes so that each box has an odd number of stones. We then proceed by placing one stone in each box to begin with, ensuring that we have a positive number in every box. Now we have <math>94</math> stones left. Because we want an odd number in each box, we pair the stones, creating <math>47</math> sets of <math>2</math>. Every time we add a pair to one of the boxes, the number of stones in the box remains odd, because (an odd #) + (an even #) = (an odd #).
 
Another way we can approach this problem is by imagining a line of 98 stones. We want to place the stones into <math>4</math> boxes so that each box has an odd number of stones. We then proceed by placing one stone in each box to begin with, ensuring that we have a positive number in every box. Now we have <math>94</math> stones left. Because we want an odd number in each box, we pair the stones, creating <math>47</math> sets of <math>2</math>. Every time we add a pair to one of the boxes, the number of stones in the box remains odd, because (an odd #) + (an even #) = (an odd #).
  
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Our problem can now be restated: how many ways are there to partition a line of <math>47</math> stones? We can easily solve this by using <math>3</math> sticks to separate the stones into <math>4</math> groups, and this is the same as arranging a line of <math>3</math> sticks and <math>47</math> stones.  <cmath>\frac{50!}{47! \cdot 3!} = 19600</cmath> <cmath>\frac{50 * 49 * 48}{3 * 2} = 19600</cmath> Out answer is therefore <math>\frac{19600}{100} = \boxed{196}</math>
+
Our problem can now be restated: how many ways are there to partition a line of <math>47</math> stones? We can easily solve this by using <math>3</math> sticks to separate the stones into <math>4</math> groups, and this is the same as arranging a line of <math>3</math> sticks and <math>47</math> stones.  <cmath>\frac{50!}{47! \cdot 3!} = 19600</cmath> <cmath>\frac{50 * 49 * 48}{3 * 2} = 19600</cmath> Our answer is therefore <math>\frac{19600}{100} = \boxed{196}</math>
  
 
== See also ==
 
== See also ==

Revision as of 12:32, 6 August 2015

Problem

Let $n$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive odd integers that satisfy $\sum_{i = 1}^4 x_i = 98.$ Find $\frac n{100}.$

Solution

Solution 1

Define $x_i = 2y_i - 1$. Then $2\left(\sum_{i = 1}^4 y_i\right) - 4 = 98$, so $\sum_{i = 1}^4 y_i = 51$.

So we want to find four integers that sum up to 51; we can imagine this as trying to split up 51 on the number line into 4 ranges. This is equivalent to trying to place 3 markers on the numbers 1 through 50; thus the answer is $n = {50\choose3} = \frac{50 * 49 * 48}{3 * 2} = 19600$, and $\frac n{100} = \boxed{196}$.

Solution 2

Another way we can approach this problem is by imagining a line of 98 stones. We want to place the stones into $4$ boxes so that each box has an odd number of stones. We then proceed by placing one stone in each box to begin with, ensuring that we have a positive number in every box. Now we have $94$ stones left. Because we want an odd number in each box, we pair the stones, creating $47$ sets of $2$. Every time we add a pair to one of the boxes, the number of stones in the box remains odd, because (an odd #) + (an even #) = (an odd #).

Our problem can now be restated: how many ways are there to partition a line of $47$ stones? We can easily solve this by using $3$ sticks to separate the stones into $4$ groups, and this is the same as arranging a line of $3$ sticks and $47$ stones. \[\frac{50!}{47! \cdot 3!} = 19600\] \[\frac{50 * 49 * 48}{3 * 2} = 19600\] Our answer is therefore $\frac{19600}{100} = \boxed{196}$

See also

1998 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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