Suppose there are 3 kinds of candy: cherry, lemon, and grape. You want to choose 8 pieces of candy by choosing an (integer) amount of each kind, which could be 0. How many ways can you do this

Answers

Answer 1

There are 9 ways to choose 8 pieces of candy from 3 kinds.

To solve this problem, we can use the concept of combinations with repetitions. We need to choose 8 pieces of candy from 3 kinds, which means we can choose 0 to 8 pieces from each kind.

We can represent this using a stars and bars diagram, where each star represents a piece of candy and the bars separate the kinds. For example, the diagram * | ** | *** represents 1 cherry, 2 lemon, and 3 grape candies.

To find the number of ways to choose 8 pieces of candy, we need to find the number of ways to arrange 8 stars and 2 bars (since there are 3 kinds of candy, there are 2 bars). This is a combination with repetition problem, and the formula is:

n + k - 1 choose k - 1

where n is the number of objects (8 stars) and k is the number of groups (2 bars). Plugging in the numbers, we get:

8 + 2 - 1 choose 2 - 1
= 9 choose 1
= 9

Therefore, there are 9 ways to choose 8 pieces of candy from 3 kinds.

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Related Questions

When you use your Taylor polynomial to estimate the probability that a value lies within two standard deviations of the mean, what do you get

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When using a Taylor polynomial to estimate the probability that a value lies within two standard deviations of the mean, the result will depend on the specific function used to create the polynomial. However, in general, a Taylor polynomial can provide a good approximation of the function within a certain interval.

1. Identify the function: The probability distribution function for a normal distribution is given by the function f(x) = (1/σ√(2π)) * e^(-(x-μ)^2 / 2σ^2), where μ is the mean and σ is the standard deviation.

2. Determine the interval: Two standard deviations from the mean are represented by the interval [μ - 2σ, μ + 2σ].

3. Apply Taylor polynomial: Approximate f(x) using a Taylor polynomial centered at μ. The higher the degree of the polynomial, the more accurate the approximation.

4. Calculate probability: Integrate the Taylor polynomial over the interval [μ - 2σ, μ + 2σ] to estimate the probability.

5. Interpret the result: The estimated probability represents the likelihood that a value lies within two standard deviations of the mean in a normal distribution.

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