Counting and Probability

1.

A vendor sells ice cream from a cart on the boardwalk. He offers vanilla, chocolate, strawberry, blueberry, and pistachio ice cream, served on either a waffle, sugar, or plain cone. How many different single-scoop ice-cream cones can you buy from this vendor?

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Your answer is :

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Answer.

\(15\)

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2.

A girl has 6 skirts, 5 blouses, and 5 pairs of shoes. How many different skirt-blouse-shoe outfits can she wear? (Assume that each item matches all the others, so she is willing to wear any combination.)

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Your answer is :

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Answer.

\(150\)

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3.

Standard automobile license plates in a country display 1 numbers, followed by 3 letters, followed by 3 numbers. How many different standard plates are possible in this system? (Assume lead 0 is not allowed, but repetitions of letters and numbers are allowed.)

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Your answer is :

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Answer.

\(158184000\)

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4.

A true-false test contains 16 questions. In how many different ways can this test be completed. (Assume we don’t care about our scores.)

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Your answer is :

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Answer.

\(65536\)

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5.

In how many ways can first, second, and third prizes be awarded in a contest with 675 contestants?

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Your answer is :

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Answer.

\(306181350\)

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6.

How many ways are there to seat 4 people in a row of 8 chairs?

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Answer.

\(1680\)

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Solution.

\({8 \choose 4} = \frac{8 !}{4 ! 4!}\)

\begin{equation*} {8 \choose 4} \cdot 4 ! =\frac{8 !}{4 ! 4!} \cdot 4! = \frac{8 !}{4!} = 1680 \end{equation*}

\(\frac{8 !}{4!} = 1680\)

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7.

An American roulette wheel has 38 slots: two slots are numbered 0 and 00, and the remaining slots are numbered from 1 to 36. Find the probability that the ball lands in an odd-numbered slot.

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Your answer is :

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Answer.

\(0.473684210526316\)

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8.

In the 6/26 lottery game, a player selects 6 numbers from 1 to 26. What is the probability of picking the 6 winning numbers?

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Your answer is :

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Answer.

\(4.34348260435217\times 10^{-6}\)

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9.

Two marbles are drawn randomly one after the other without replacement from a jar that contains 2 red marbles, 7 white marbles, and 3 yellow marbles. Find the probability of the following events.

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(a) A red marble is drawn first followed by a white marble.

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The probability is :

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(b) A white marble is drawn first followed by a white marble.

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The probability is :

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(c) A yellow marble is not drawn at all.

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The probability is :