Tossing 12 different valued coins at the same time

Question

I toss $3$ dimes, $4$ nickels, and $5$ pennies all at the same time. What is the chance that all of the ones that land heads up is $30$ cents?

This is from a timed competition, fastest answers are the best.

My answer: The denominator should be $2^{12}$ since we are throwing $12$ coins. There are $5$ cases of getting $30$ cents.

1. $3$ dimes
2. $2$ dimes, $2$ nickels
3. $2$ dimes, $1$ nickels, $5$ pennies
4. $1$ dimes, $4$ nickels
5. $1$ dimes, $3$ nickels, $5$ pennies

For #1, there is only one option

For #2, there is $3\choose 2$ $\cdot$ $4\choose 2$ $= 18$, since we are picking $2$ out of $3$ dimes and $2$ out of $4$ nickels.

For #3, it would be $ 3 \cdot 4 = 12$, since we are picking $2$ out of $3$ dimes and $1$ out of $4$ nickels.

For #4, it would be be $3$

For #5, it would be $ 3 \cdot 4 = 12$.

My final answer is$\frac{46}{2^{12}}$

I'm not sure this is 100% correct, and this definitely isn't the fastest way. Can anyone check if I'm correct, and if not, tell me what is wrong? Faster answers is greatly appreciated.

Answer 1

Let's see if a simulation gives a close approximation to your answer. A million plays of the game should give probabilities accurate to about 2 or 3 places.

set.seed(2020)
p = rbinom(10^6,5,.5)
n = 5*rbinom(10^6,4,.5)
d = 10*rbinom(10^7,3,.5)
t = p + n + d
mean(t == 30)
[1] 0.0111551    # aprx P(T = 30)
2*sd(t==30)/1000
[1] 0.0002100539 # 95% margin of sim error
46/2^12
[1] 0.01123047   # Your answer
mean(t >= 30)
[1] 0.4125161

The simulated result $P(T = 30) = 0.0112\pm 0.0002$ seems to match your answer. Also, $P(T \ge 30) \approx 0.413.$

Here is a histogram with simulated probabilities of the distribution of $T.$

hist(t, prob=T, br=(0:56)-.5, col="skyblue2")
abline(h=46/2^12, col="orange")

Note: In R the vector t has a million totals. The vector t==30 is a logical vector of a million TRUEs and FALSEs. Its mean is the proportion of its TRUEs.