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J is a collection of four odd integers and the greatest [#permalink]

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30 Aug 2006, 02:28

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J is a collection of four odd integers and the greatest difference between any two integers in J is 4. The standard deviation of J must be one of how many numbers?

(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

Last edited by kevincan on 30 Aug 2006, 04:13, edited 1 time in total.

Interesting.
Will give Ex. cause it will be easier to explain.
Diff-0 3,3,3,3
Diff-2 and 0 3,3,3,5
Diff-4 and 0 3,3,3,7
Diff-2 and 4and 0 3,3,5,7
Diff 2 and 0 3,3,5,5
diff 4 and 0 3,3,7,7
Difference can not be 1 and 3 cause it will give even integer
So should be D)

Let four numbers are a,b,c,d and are in increasing order. d-a = 4
So a and d are fixed. We can change b and c but there is only one number between a and d. Let that number be z

There are six possibilities.
a,a,a,d
a,d,d,d
a,z,z,d
a,a,z,d
a,z,d,d
a,a,d,d

Let four numbers are a,b,c,d and are in increasing order. d-a = 4 So a and d are fixed. We can change b and c but there is only one number between a and d. Let that number be z

There are six possibilities. a,a,a,d a,d,d,d a,z,z,d a,a,z,d a,z,d,d a,a,d,d

Hence answer is 6.

Won't some of these have the same s.d.?

I know that two of these will have same mean but I thought the dispersion will be different.
_________________

Let four numbers are a,b,c,d and are in increasing order. d-a = 4 So a and d are fixed. We can change b and c but there is only one number between a and d. Let that number be z

There are six possibilities. a,a,a,d a,d,d,d a,z,z,d a,a,z,d a,z,d,d a,a,d,d

Hence answer is 6.

I think 7. The other possibility is where the standard deviation is 0
aaaa or dddd or zzzz

Let four numbers are a,b,c,d and are in increasing order. d-a = 4 So a and d are fixed. We can change b and c but there is only one number between a and d. Let that number be z

There are six possibilities. a,a,a,d a,d,d,d a,z,z,d a,a,z,d a,z,d,d a,a,d,d

Hence answer is 6.

Agree with this explanation... D should be the answer.
_________________

Uh uh. I know what you're thinking. "Is the answer A, B, C, D or E?" Well to tell you the truth in all this excitement I kinda lost track myself. But you've gotta ask yourself one question: "Do I feel lucky?" Well, do ya, punk?

Hmmm - does that mean that the max difference can be 0, 2, or 4?

Because then we get -

a, a, z, d
a, z, z, d
a, z, d, d
a, a, d, d
a, d, d, d
a, a, a, d
a, z, z, z
a, a, a, z
a, a, z, z
z, z, z, d
z, d, d, d
z, z, d, d

SD = 0 for
a, a, a, a
z, z, z, z
d, d, d, d

Too many possible SDs (for the choices given)?

I think we need to assume that there always exists a pair in the set, for which difference is 4. The remaining choices -

a, a, z, d
a, z, z, d
a, z, d, d
a, a, d, d
a, d, d, d
a, a, a, d

6 SDs...
_________________

Uh uh. I know what you're thinking. "Is the answer A, B, C, D or E?" Well to tell you the truth in all this excitement I kinda lost track myself. But you've gotta ask yourself one question: "Do I feel lucky?" Well, do ya, punk?

I started with the premise that 4 IS the greatest difference, but there are two elements in the set with a difference of 4.

(so no aaaa, dddd etc sets)

Now there are only three elements in this set, or two, because of the constraints, e.g. if the two numbers are 5 and 9, then at most the other member of hte set can be 7.

Thus the members can be:
5559
5999
(same std dev, diff mean)
5599
5779
(same std dev, same mean)
5579
5799
(same std dev, diff means)

Thus, provided I havent missed anything, the max std dev should be 3. Interestingly, the max means can be only 5. Not 6.

Basically we need to find out the different SDs we can get.

If x = n.SD^2 is unique, then SD must be unique

(1111): M = 1, X = 0
(1113): M = 1.5, X = 0.25+0.25+0.25+2.25 = 3
(1115): M = 2, X = 1+1+1+9 =12
(1135): M = 2.5, X = 2.25+2.25+0.25+6.25 = 11
(1133): M = 2, X = 1+1+1+1 = 4
(1155): M = 4, X = 4+4+4+4 = 16
(1355): M = 3.5, X = 6.25+0.25+2.25+2.25 = 11
(1335): M = 3, X = 4+4 = 8

From the above, we have X = {0,3,4,8,11,11,12,16} which gives us 8 values. Of these, 7 are distinct. Hence I choose E.

J is a collection of four odd integers and the greatest difference between any two integers in J is 4. The standard deviation of J must be one of how many numbers?

(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

i found 4.

lets say the integers are 1, 3 and 5. the four integers can be:

1 1 3 5
1 1 1 5
1 1 5 5
1 3 3 5
1 3 5 5
1 5 5 5

but two pairs have the same SD. therefore finally there are 4 different SDs.

is it true that we must have 1 and 5 in J
Because the quest just say the greatest diff between any two number = 4. It doesn't mean that there must be 2 num in J which their diff =4
I think this is a too tough quest and it wastes too much time.