There are 300 cars in Bill Gates driveway. Bill decided to paint a third of the cars to pink and another third of the cars blue.
(a) How many ways are there to paint Bill's cars? Answer should be given in terms of factorials.
(b) Use Stirling formula to approximate your answer in (a).
Im trying to get understand these types of problems but my textbook is hard to understand. Thanks for your help.
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Use the multinomial selection: 100 cars to be blue, 100 to be pink, and 100 not to be painted from 300.
$$ \binom{300}{100,100,100}= \frac{300!}{100!^3}$$
Use Stirling's Approximation: $n!\approx \sqrt{2\pi n}\left(\frac n e\right)^n$
$$\frac{300!}{100!^3} \approx \frac{3^{300}\sqrt{3}}{200\pi}$$
So Bill gates is painting $100$ cars pink, and another $100$ blue. Let's start by picking the pink cars. There are $\binom{300}{100}$ ways to choose the pink cars. We then need to choose the blue cars. There are $200$ cars from which to choose, so we have $\binom{200}{100}$ ways to choose the blue cars. So by rule of product, we multiply these quantities: $\binom{200}{100} * \binom{300}{100}$.