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# A jewelry maker chooses three beads at random from a bag with 10 beads labeled A, B, C, D, E. F, G, H, I. and J. a. How can you use permutations or combinations to find P(selected beads spell the initials DEB)? What is the probability? b. How can you use permutations or combinations to find P(selected beads are all vowels)? What is the probability?

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I am assuming that it they are asking about 1st bead is D, 2nd is E, 3rd is B, and not that D,E,B are drawn in any order so could be arranged to spell DEB. It seems to be too much to ask for Pearson to be clear.

a) \ Total ways to choose 3 beads from 10 where order is important (permutation):

\begin{align*} _nP_r &= \dfrac{ n!}{ (n-r)! } \\\\ _{10}P_3 &= \dfrac{10!}{ (10-3)! } = 10(9)(8) = 720 \end{align*}

There is only one way for it to be spelled correctly, 1, so the probability is

\begin{align*} P(\text{spells DEB} ) &= \dfrac{1}{720} \approx 0.0014 \qquad (0.14\%) \end{align*}

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