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

Hi All,

Please help me out with this problem..it's eating me for a long time now

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
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June 18th, 2009, 08:09 PM   #2
nqh
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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.
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December 2nd, 2009, 01:41 PM   #3
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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.



And you are interested in the coefficient of x^14, that will be tha answer. Expanding that expression we get:




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.
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December 6th, 2009, 04:53 PM   #4
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Moved from Math Olympiads forum.
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