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My question is about a method to approach counting in MATHCOUNTS States #29 2019 by using generating functions.

Here is the problem:

Chris flips a coin 16 times. Given that exactly 12 of the flips land heads, what is the probability that Chris never flips five heads in a row? Express your answer as a common fraction.

Now, there are $\dbinom{12}4$ possible combinations, but the counting at the top is giving me a problem.

To turn this problem into a generating function problem, I considered the amount of heads before each tail. $0,1,2,3,$ or $4$ heads are allowed. Now, we have to do this $5$ times - $4$ before a coin and $1$ at the end. Therefore, we need the coefficient of $x^{12}$ in $(x^4+x^3+x^2+x+1)^5$, which would be our number. However, from here, I cannot do much to simplify this. I can try expressing it as $\frac{(x-1)^5}{(x^5-1)^5}$ but I can't follow with any combinatorics.

I came here to ask if there is a method in which this generating function can be applied which does not involve major computation. (i.e. expanding)

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  • $\begingroup$ Your generating function is upside down. $\endgroup$ – arctic tern May 18 at 7:03
  • $\begingroup$ Adding to the previous comment: to be precise, $1+x+x^2+x^3+x^4=\frac{1-x^5}{1-x}$ (or $\frac{x^5-1}{x-1}$ if you prefer). $\endgroup$ – Marc van Leeuwen May 18 at 7:05
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Write it as $(1-x^5)^5(1-x)^{-5}$. One may apply the binomial expansion to the first power, and the newton-binomial expansion to the second (even when the exponent $-5$ is negative). One gets

$$ \left[\binom{5}{0}-\binom{5}{1}x^5+\binom{5}{2}x^{10}-\cdots\right]\left[1-\binom{-5}{1}x+\binom{-5}{2}x^2-\cdots\right]. $$

Therefore the coefficient of $x^{12}$ is given by combining like terms:

$$ \binom{5}{0}\binom{-5}{12}-\binom{5}{1}\binom{-5}{7}+\binom{5}{2}\binom{-5}{2}. $$

Note you can evaluate binomial coefficients with negative top argument via

$$ \binom{n}{k}:=\frac{n(n-1)\cdots}{k(k-1)\cdots} $$

(There are $k$ factors up top and down below.)

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  • $\begingroup$ +1. Alternatively to working out the negative upper index binomial coefficients, one may use the negation formula $\binom{-n}k=(-1)^k\binom{n+k-1}k$, so your answer becomes $\binom50\binom{16}{12}+\binom51\binom{11}7+\binom52\binom62=3620$ $\endgroup$ – Marc van Leeuwen May 18 at 7:11

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