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<blockquote data-quote="frisbeet" data-source="post: 768293" data-attributes="member: 10287">An example, because I don't want this thread to die today.If# ways = COMB(oc-1,n-1) + SUM(i=1,n-1) {(-1)^i * IF[oc&lt;(i*s+n),0,COMB(n,i)*COMB(oc-i*s-1,n-1)]}Thenfor 4d6,# ways of getting a 17 == COMB(16,3) - COMB(4,1)*COMB(10,3) + COMB(4,2)*COMB(4,3) - 0= 16!/(13!*3!) - 4!/(3!*1!)*10!/(7!*3!) + 4!/(2!*2!)*4!/(1!*3!)= 16*15*14/6 - 4*10*9*8/6 + (4*3/2)*4= 560 - 480 + 24= 104Sure enough, there are 104 ways to sum to 17 with 4 six-sided dice.&nbsp; Here's how I counted it.&nbsp; Determine all possible outcomes, then how many ways there are of rolling each outcome.&nbsp; So that's:oc (each digit is a single die result)__# ways6641___126632___126551___126542___246533___126443___125552___45543___125444___4sum of # ways: 104.&nbsp; Not a proof but good enough for me.EDIT:changed= COMB(17,3) - COMB(4,1)*COMB(10,3) + COMB(4,2)*COMB(4,3) - 0to = COMB(16,3) - COMB(4,1)*COMB(10,3) + COMB(4,2)*COMB(4,3) - 0</blockquote>

[QUOTE="frisbeet, post: 768293, member: 10287"] An example, because I don't want this thread to die today. If # ways = COMB(oc-1,n-1) + SUM(i=1,n-1) {(-1)^i * IF[oc<(i*s+n),0,COMB(n,i)*COMB(oc-i*s-1,n-1)]} Then for 4d6, # ways of getting a 17 = = COMB(16,3) - COMB(4,1)*COMB(10,3) + COMB(4,2)*COMB(4,3) - 0 = 16!/(13!*3!) - 4!/(3!*1!)*10!/(7!*3!) + 4!/(2!*2!)*4!/(1!*3!) = 16*15*14/6 - 4*10*9*8/6 + (4*3/2)*4 = 560 - 480 + 24 = 104 Sure enough, there are 104 ways to sum to 17 with 4 six-sided dice. Here's how I counted it. Determine all possible outcomes, then how many ways there are of rolling each outcome. So that's: oc (each digit is a single die result)__# ways 6641___12 6632___12 6551___12 6542___24 6533___12 6443___12 5552___4 5543___12 5444___4 sum of # ways: 104. Not a proof but good enough for me. EDIT: changed = COMB(17,3) - COMB(4,1)*COMB(10,3) + COMB(4,2)*COMB(4,3) - 0 to = COMB(16,3) - COMB(4,1)*COMB(10,3) + COMB(4,2)*COMB(4,3) - 0 [/QUOTE]

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