Drawing 7 cards, what is the probability 3 will be hearts?

Question

Problem:

We select $7$ playing cards out of $52$. What is the probability that among the selected cards there are exactly three hearts?
Answer:

Let $p$ be the probability we seek. \begin{align*} p &= \dfrac{ { {13} \choose {3} } { {39} \choose {4} } } { { {52} \choose {7} } } \\ {{13} \choose {3}} &= \dfrac{ 13(12)(11) }{ 3(2)} = 13(2)(11) \\ {{13} \choose {3}} &= 286 \\ {{39} \choose {4} } &= \dfrac{39(38)(37)(36) }{ 4(3)(2) } \\ {{39} \choose {4} } &= \dfrac{39(19)(37)(36) }{ 4(3) } \\ {{39} \choose {4} } &= 39(19)(37)(3) = 82251 \\ {52} \choose {7} &= \dfrac{ 52(51)(50)(49)(48)(47)(46) }{ 7(6)(5)(4)(3)(2)} \\ {52} \choose {7} &= \dfrac{ 52(51)(50)(49)(48)(47)(23) }{ 7(6)(5)(4)(3)} \\ {52} \choose {7} &= \dfrac{ 52(51)(50)(49)(16)(47)(23) }{ 7(6)(5)(4) }\\ {52} \choose {7} &= \dfrac{ 52(51)(50)(49)(4)(47)(23) }{ 7(6)(5) }\\ {52} \choose {7} &= \dfrac{ 52(51)(10)(49)(4)(47)(23) }{ 7(6) }\\ {52} \choose {7} &= \dfrac{ 52(51)(5)(49)(4)(47)(23) }{ 7(3) }\\ {52} \choose {7} &= \dfrac{ 52(17)(5)(49)(4)(47)(23) }{ 7 }\\ {52} \choose {7} &= 52(17)(5)(7)(4)(47)(23) \\ p &= \dfrac{ 286(82251) }{ 52(17)(5)(7)(4)(47)(23) } \\ \end{align*} Now we need to simplify the fraction to find $p$. \begin{align*} p &= \dfrac{ 143(82251) }{ 52(17)(5)(7)(2)(47)(23) } \\ p &\doteq 0.176 \end{align*} However, the website where I got the problem has the answer: $$ 17.7\% $$ Where did I go wrong?

Answer 1

I believe the post is correct because of the following session in SciLab:

(nchoosek(13,3)*nchoosek(39,4))/ nchoosek(52,7)
ans =

0.1758333

Answer 2

Let H = draw a heart and N = not drawing a heart. The possibilities are HHHNNNN, NHHHNNN, NNHHHNN, NNNHHHN, NNNNHHH, HNHHNNN, HHNHNNN, etc. (just make an organized list of all possibilities). Then calculate the probability of each outcome (the first is $\frac{1}{4} \times \frac{12}{51} \times \frac{11}{50} \times \frac{39}{49} \times \frac{38}{48} \times \frac{37}{47} \times \frac{36}{46}$ ) and add them together. But it looks like your solution is correct and the computer doesn't recognize that it has a rounded answer.