My Math Forum Problem related to Cricket (Permutation & Combination)

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March 26th, 2008, 12:25 AM   #1
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Problem related to Cricket (Permutation & Combination)

Hi All,

Quote:
 In how many ways a batsman can score 14 runs in 6 balls not scoring more than 4 runs in any ball.
Thanks for trying.

Satya

 June 18th, 2009, 08:09 PM #2 Newbie   Joined: Jun 2009 Posts: 9 Thanks: 0 Re: Problem related to Cricket (Permutation & Combination) We have equation: X1 + X2 + X3 + X4 + X5 + X6 = 14 with condition 0 ? X1, X2, X3, X4, X5, X6 ? 4 and Xi integer (i = 1 -> 6) The total roots of above equation is the ways a batsman can score 14 runs The above equation have 1506 roots as below: 0 0 2 4 4 4 0 0 3 3 4 4 0 0 3 4 3 4 0 0 3 4 4 3 0 0 4 2 4 4 0 0 4 3 3 4 0 0 4 3 4 3 0 0 4 4 2 4 0 0 4 4 3 3 0 0 4 4 4 2 0 1 1 4 4 4 0 1 2 3 4 4 0 1 2 4 3 4 ....... 4 4 2 4 0 0 4 4 3 0 0 3 4 4 3 0 1 2 4 4 3 0 2 1 4 4 3 0 3 0 4 4 3 1 0 2 4 4 3 1 1 1 4 4 3 1 2 0 4 4 3 2 0 1 4 4 3 2 1 0 4 4 3 3 0 0 4 4 4 0 0 2 4 4 4 0 1 1 4 4 4 0 2 0 4 4 4 1 0 1 4 4 4 1 1 0 4 4 4 2 0 0 (I can't post a full list of roots because very long. If you want I will send by attachment file) Conclusion have 1506 ways a batsman can score 14 runs in 6 balls not scoring more than 4 runs in any ball.
 December 2nd, 2009, 01:41 PM #3 Senior Member   Joined: Dec 2009 From: Las Vegas Posts: 209 Thanks: 0 Re: Problem related to Cricket (Permutation & Combination) Hi satyabrata_pati; The easiest way to solve that problem is to use an ordinary generating function. For the conditions a+b+c+d+e+f = 14 where 0 <= a,b,c,d,e,f <=4 you have the generating function. $(1 + x +x^2 +x^3 + x^4)^6$ And you are interested in the coefficient of x^14, that will be tha answer. Expanding that expression we get: $(1 + x +x^2 +x^3 + x^4)^6=x^{24}+6 x^{23}+21 x^{22}+56 x^{21}+126 x^{20}+246 x^{19}+426 x^{18}+666 x^{17}+951 x^{16}+1246 x^{15}+1506 x^{14} \\ \hspace{180} +1686 x^{13}+1751 x^{12}+1686 x^{11}+1506 x^{10}+1246 x^9+951 x^8+666 x^7+426 x^6+246 x^5+126 x^4+56 x^3 \\ \hspace{180} +21 x^2+6 x+1$ The coefficient for x^14 is 1506 so there are 1506 ways. The nice thing about gf's is that they solve the problem entirely. To see how many ways to get 20 runs just check the coeff. of x^20.
 December 6th, 2009, 04:53 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,746 Thanks: 2133 Moved from Math Olympiads forum.

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