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Jen teaches a ceramics class on the weekends. In Sunday's class, the students created $9$ clay objects, $7$ of which were picture frames. After each class, Jen bakes the clay objects in a hot kiln to harden them. Students pick up their finished pieces later in the week. $\newline$ If Jen randomly chose $6$ clay objects to bake in the kiln Sunday night, what is the probability that all of them are picture frames? $\newline$ Write your answer as a decimal rounded to four decimal places._________

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Find probabilities using combinations and permutations

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Q. Jen teaches a ceramics class on the weekends. In Sunday's class, the students created

9

clay objects,

7

of which were picture frames. After each class, Jen bakes the clay objects in a hot kiln to harden them. Students pick up their finished pieces later in the week.

\newline

If Jen randomly chose

6

clay objects to bake in the kiln Sunday night, what is the probability that all of them are picture frames?

\newline

Write your answer as a decimal rounded to four decimal places._________

Calculate Total Ways: First, let's figure out the total number of ways to choose $6$ objects out of $9$ without worrying about what type they are. We use the combination formula for this, which is $C(n, k) = \frac{n!}{k!(n-k)!}$ , where $n$ is the total number of objects and $k$ is the number of objects we want to choose.
Choose $6$ Picture Frames: Calculate the total number of ways to choose $6$ objects from $9$ : $C(9, 6) = \frac{9!}{6!(9-6)!} = \frac{9!}{6!3!} = \frac{9\times8\times7}{3\times2\times1} = 84$ .
Calculate Probability: Now, we need to find the number of ways to choose $6$ picture frames from the $7$ available. Again, we use the combination formula: $C(7, 6) = \frac{7!}{(6!(7-6)!)} = \frac{7!}{(6!1!)} = \frac{7}{1} = 7$ .
Calculate Probability: Now, we need to find the number of ways to choose $6$ picture frames from the $7$ available. Again, we use the combination formula: $C(7, 6) = \frac{7!}{(6!(7-6)!)} = \frac{7!}{(6!1!)} = \frac{7}{1} = 7$ .To find the probability that all $6$ chosen objects are picture frames, we divide the number of ways to choose $6$ picture frames by the total number of ways to choose $6$ objects: Probability = $\frac{C(7, 6)}{C(9, 6)}$ .
Calculate Probability: Now, we need to find the number of ways to choose $6$ picture frames from the $7$ available. Again, we use the combination formula: $C(7, 6) = \frac{7!}{(6!(7-6)!)} = \frac{7!}{(6!1!)} = \frac{7}{1} = 7$ .To find the probability that all $6$ chosen objects are picture frames, we divide the number of ways to choose $6$ picture frames by the total number of ways to choose $6$ objects: Probability = $\frac{C(7, 6)}{C(9, 6)}$ .Calculate the probability: Probability = $\frac{7}{84} = \frac{1}{12}$ . To convert this to a decimal, we divide $1$ by $12$ .

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