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A batsman can score $0,2,3$ or $4$ runs for each ball he receives.If $N$ is the number of ways of scoring a total of 20 runs in one over of six balls.Then find $N$.


Different options of scoring $20$ runs in one over are
$(1)2,2,4,4,4,4$
$(2)0,4,4,4,4,4$
$(3)3,3,4,4,4,4$
There are $\frac{6!}{2!4!}=15$ ways for (1) and (3) and $6$ ways for (2).That adds to $36$.But the book answer says total ways of scoring $20$ runs are $96$.I dont know where have i gone wrong.Please help me.

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  • $\begingroup$ Whoever wrote that question wasn’t thinking: if he scores $3$, he’ll no longer be taking strike, so he won’t receive all six balls of the over! $\endgroup$ – Brian M. Scott Oct 8 '15 at 14:48
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You forgot these:

(4) 3,3,3,3,4,4

(5) 2,3,3,4,4,4

And (3) does not add up to 20.

So rewriting it gives you:

(1)2,2,4,4,4,4 =15

(2)0,4,4,4,4,4 =6

(3) 3,3,3,3,4,4 =15

(4) 2,3,3,4,4,4 =60

Adding up to 96!

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Solve this using Multinomial Theorem. an instance of a ball can be scored for either 0,2,3,4,6 runs and there are 6 such instances. So, now we have an event which can have the following exponents - 0,2,3,4,6 and there are 6 occurrences of the event. The expression is given as follows

(x^0+x^2+x^3+x^4+x^6)^6 

The task is to find the exponent of x^20. Try to solve it this way. Comment if you are struck.

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