The original problem is:
"You choose a letter at random from the word Mississippi eleven times without replacement. What is the probability that you can form the word Mississippi with the eleven chosen letters? Hint: it may be helpful to number the eleven letters as 1,2,…,11."
I've looked through other pages which tell me the answer should be: 2.88*10^-5
Yet my book tells me it's ${11 \choose 4}*{7 \choose 4}*{3 \choose 2}*4^4*4^4*2^2*(1/(11^{11}))$ = .0318…
Is my book wrong? How would you approach this question?
Best Answer
Your book is essentially asking what is the probability that picking $11$ letters from Mississippi WITH replacement gives you $11$ letters that can be rearranged to form Mississippi.
I agree with your interpretation, and the way you did it is absolutely consistent, but that's how the book (badly) interprets it.
One way is to simply this is first find the probability that you can get SSSSIIIIPPM which is $\left(\frac{4}{11}\right)^4\left(\frac{4}{11}\right)^4\left(\frac{2}{11}\right)^2\left(\frac{1}{11}\right)^1$. That can be rearranged into MISSISSIPPI. There are $\frac{11!}{4!4!2!1!}$ different permutations of MISSISSIPPI. So we have to find the probability of getting any one of those permutations, which is a $\frac{11!}{4!4!2!1!}\left(\frac{4}{11}\right)^4\left(\frac{4}{11}\right)^4\left(\frac{2}{11}\right)^2\left(\frac{1}{11}\right)^1 = \boxed{0.031837}$ chance.