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(a) Count the number of quadruples (a, b, c, d) of nonnegative integers such that 0<= a < b < c < d <= 12.

(b) For this part, we want to count the number of quadruples (a, b, c, d) of nonnegative integers such that  0 <= a <= b <= c <= d <= 12.
Here, some of  a, b, c and d can be equal to each other, so the answer will be different from part (a). Each value a, b, c, d must be between 0 and 12 inclusive. One idea is to count how many times each number appears.


(c) In general, find the number of  k-tuples (a_1, a_2, ... a_k) of nonnegative integers such that
 0 <= a_1 <= a_2 <= a_3 <= . . . <= a_k <= n.
 

 Apr 7, 2020
 #1
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The question above has already been answered here.

 Apr 7, 2020
 #4
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I checked the answer to that question and the answerer did not consider 0 as a valid digit, therefore all of his/her answers were based off of 12 instead of 13.

DeSTrOYer303  Apr 7, 2020
 #5
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Yes, it is 13 C 4.... 0 and 12 are included...

tertre  Apr 7, 2020
 #6
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Yeah I just saw that...

HELPMEEEEEEEEEEEEE  Apr 7, 2020
 #2
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(a): This seems complicated, yet I think 13 C 4 = 715 ways if a is the least, b is the second to least, etc.

 

Yes, you're right...

 Apr 7, 2020
edited by tertre  Apr 7, 2020
 #3
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+1

There would be 13 integers to choose from right?

 

So it would be 13 C 4= 715

HELPMEEEEEEEEEEEEE  Apr 7, 2020
edited by HELPMEEEEEEEEEEEEE  Apr 7, 2020
edited by HELPMEEEEEEEEEEEEE  Apr 7, 2020
 #7
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I did get 715 for the first part, and I did not have trouble with that. For part b, I just don't know whether to approach this problem with casework or some sort of counting technique like stars and bars. One of the hints for part b was to approach the problem by designating variables.

DeSTrOYer303  Apr 7, 2020
edited by DeSTrOYer303  Apr 7, 2020
 #8
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For b., if a were 13 then there is one arrangement where all of them are 13, if a were 12 then there are 4 arragements... I wonder if there is a pattern.

HELPMEEEEEEEEEEEEE  Apr 7, 2020
 #9
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There are 10 arrangements if a is 11...

HELPMEEEEEEEEEEEEE  Apr 7, 2020
 #10
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Is there 20 arrangements if a is 10? 

DeSTrOYer303  Apr 7, 2020
 #11
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I'm not seeeing any pattern here, so we should probably look for something else...

HELPMEEEEEEEEEEEEE  Apr 7, 2020
 #12
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I mean the numbers are in order of 3 C 3, 4 C 3, 5 C 3...

DeSTrOYer303  Apr 7, 2020
 #13
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That is a great observation! If that is the pattern, then you should be able to solve the problem from there!

 Apr 7, 2020
 #14
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Thanks!

DeSTrOYer303  Apr 7, 2020

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