Mapping Values to 14 categories (Math and permutations stuff) - php

Question

My app has people put 24 groups of 4 statements in order. In each group of 4 one is the "D" statement, one is the "I" statement, one is the "S" statement, and one is the "C" statement.
So the end result looks something like ['ISCD','CISD','DISC',CISD,'CISD','ISCD'...] because the are essentially rearranging the 4 letters
In the end, they get a "score" for each letter using the following algorithm.
For each of I,S,C and D
Find the number of times that letter is first and multiply by 3
Find the number of times that letter is second and multiply by 2
Find the number of times that letter is third and muliply by 1
Total it up, and that is the score for that letter
The end result is that each letter (I,S,D,C) gets a score from 0 to 72, and there are always 144 total points given out:
I want to map the results to 14 reports:
D
I
S
C
DI
IS
SC
CD
DS
IC
DIS
ISC
SCD
CDI
The idea is that if S is dominant, we choose the S report. If Both D and I are dominant, we choose the DI report. If none is particularly dominant, we choose the top 3. (there is no difference between DI and ID meaning which one is most dominant is irrelevant if they are both high)
So if the scores are D=50, I=48, S=20,C=26 then I want it to choose "DI" since D and I are dominant. There are 24^(4!) possible responses from the user, that I need to map to 14 reports
I understand that I will have to set the thresholds for what "dominant" means, but for starters, I want to assume all possible responses are equally likely, and to map all possible responses to the 14 reports to where each of the 14 reports is equally likely, given random input.
I expect it's 1 to 5 lines of code. It'll be in php but any language including math or pseudo code should be fine.
UPDATE:
I figured out a way to do it in one line of code, but it's not evenly distributed. here's the php (no dependencies)
<?php
$totals=array();
$lets=array('D','I','S','C');
for($j=0;$j<100000;$j++)
{
$vals=array('D'=>0,'I'=>0,'S'=>0,'C'=>0);
for($i=0;$i<24;$i++)
{
shuffle($lets);
$vals[$lets[0]]+=3;
$vals[$lets[1]]+=2;
$vals[$lets[2]]+=1;
}
$D=$vals['D'];$I=$vals['I'];$S=$vals['S'];$C=$vals['C'];
//calculate which report
$reportKey=($D>36?'D':'').($I>36?'I':'').($S>36?'S':'').($C>36?'C':'');
if(!$reportKey)
$reportKey="DIS";
if(isset($totals[$reportKey]))
$totals[$reportKey]+=1;
else
$totals[$reportKey]=1;
echo $reportKey." $D $I $S $C <br>";
}
echo "<br>";
foreach ($totals as $k=>$v)
echo "$k: $v<br>";
The magic line is
$reportKey=($D>36?'D':'').($I>36?'I':'').($S>36?'S':'').($C>36?'C':'');
That line says if any value is over 36, include that letter. the output of the script is like this:
SC 35 33 38 38
IC 33 42 32 37
DI 44 39 29 32
...
...
DC 46 21 35 42
DIS 38 37 40 29
IC 36 39 28 41
DS 41 36 42 25
C 36 34 29 45
IS 29 41 38 36
IS 28 46 41 29
DS 38 33 40 33
DS 41 33 40 30
DS: 1444
D: 889
IS: 1466
S: 910
C: 874
SC: 1442
IC: 1467
DI: 1569
ISC: 407
DSC: 386
DIS: 388
DC: 1487
DIC: 396
I: 875
As you can see, it automatically split it into 14 categories, but the distribution varies with the 2 letter ones being way more likely.

Answer 1

You can do this recursively using Haskell e.g. as follows:
combinationsOf _ 0 = [[]]
combinationsOf [] _ = []
combinationsOf (x:xs) k = map (x:) (combinationsOf xs (k-1) ) ++ combinationsOf xs k
The results from GHCI:
*Main> concatMap (combinationsOf "DISC") [1,2,3]
["D","I","S","C","DI","DS","DC","IS","IC","SC","DIS","DIC","DSC","ISC"]