Question

1. donations emily has 20 collectible dolls from different countries that she will donate. if she selects 10 of them at random, what is the probability that she chooses the dolls from ecuador, paraguay, chile, france, spain, sweden, switzerland, germany, greece, and italy?

Explanation:

Step1: Determine total number of ways to choose 10 dolls from 20.

The formula for combinations is \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 20 \) and \( k=10 \). So \( C(20,10)=\frac{20!}{10!(20 - 10)!}=\frac{20!}{10!×10!} \).

Step2: Determine number of favorable outcomes.

We need to choose 10 dolls that include 9 specific dolls (Ecuador, Paraguay, Chile, France, Spain, Sweden, Switzerland, Germany, Greece, Italy? Wait, wait, the 9 countries? Wait, no, wait: the problem says "chooses the dolls from Ecuador, Paraguay, Chile, France, Spain, Sweden, Switzerland, Germany, and Italy" – that's 9 dolls. So we need to choose 10 dolls, and 9 of them are fixed (these 9), so we need to choose 1 more doll from the remaining \( 20 - 9=11 \) dolls. So the number of favorable combinations is \( C(11,1)=\frac{11!}{1!(11 - 1)!}=\frac{11!}{1!×10!}=11 \).

Step3: Calculate the probability.

Probability \( P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{C(11,1)}{C(20,10)} \).

First, calculate \( C(20,10)=\frac{20×19×18×17×16×15×14×13×12×11}{10×9×8×7×6×5×4×3×2×1}=184756 \).

\( C(11,1) = 11 \).

So \( P=\frac{11}{184756}=\frac{1}{16796} \) (Wait, 184756 ÷ 11 = 16796? Let's check: 11×16796 = 11×(16000 + 796)=176000+8756 = 184756. Yes. So 11/184756 = 1/16796? Wait, no: 11 divided by 184756: let's simplify. 184756 ÷ 11 = 16796, so 11/184756 = 1/16796? Wait, no, 11 and 184756: GCD of 11 and 184756. 184756 ÷ 11 = 16796, so yes, 11/184756 = 1/16796? Wait, no, 11 is the numerator, 184756 is the denominator. 184756 ÷ 11 = 16796, so 11/184756 = 1/16796. Wait, but let's re - check the favorable outcomes. Wait, the problem says "chooses the dolls from Ecuador, Paraguay, Chile, France, Spain, Sweden, Switzerland, Germany, and Italy" – that's 9 dolls. She is choosing 10 dolls, so she must include these 9 and 1 more from the remaining 20 - 9 = 11. So the number of favorable is C(11,1)=11. Total number of ways to choose 10 from 20 is C(20,10)=184756. So probability is 11/184756 = 1/16796 ≈ 5.95×10^{-5}? Wait, no, 1/16796 is approximately 0.0000595, or 5.95×10^{-5}. But let's check the combination calculation again.

Wait, \( C(20,10)=\frac{20!}{10!10!}=\frac{20×19×18×17×16×15×14×13×12×11}{10×9×8×7×6×5×4×3×2×1} \). Let's compute numerator: 20×19 = 380; 380×18 = 6840; 6840×17 = 116280; 116280×16 = 1860480; 1860480×15 = 27907200; 27907200×14 = 390700800; 390700800×13 = 5079110400; 5079110400×12 = 60949324800; 60949324800×11 = 670442572800. Denominator: 10!×10! = (3628800)×(3628800)=13168189440000. Wait, no, that's wrong. Wait, 10! = 3628800. So 10!×10! = 3628800×3628800. But when we calculate \( C(20,10) \), we can use the formula \( C(n,k)=\frac{n!}{k!(n - k)!} \), and for n = 20, k = 10, \( C(20,10)=184756 \) (this is a known value, the number of ways to choose 10 elements from 20 is 184756). And \( C(11,1)=11 \). So the probability is 11/184756 = 1/16796 ≈ 0.0000595, or 5.95×10^{-5}.

Answer:

The probability is \(\frac{1}{16796}\) (or approximately \(5.95\times10^{-5}\)).