8. How many different mixes of candy are possible if a mix consists of 10 pieces of candy and 4 different kinds of candy are available in unlimited quantities?
8. How many different mixes of candy are possible if a mix consists of 10 pieces of candy and 4 different kinds of candy are available in unlimited quantities?
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Question 8: Combinatorial Candy Mixes**
How many different mixes of candy are possible if a mix consists of 10 pieces of candy and 4 different kinds of candy are available in unlimited quantities?
This problem can be solved using the "stars and bars" theorem in combinatorics. It involves calculating the number of ways to distribute a fixed number of identical items (in this case, 10 candies) into a fixed number of distinct categories (4 types of candy).
**Solution Approach**
Using the stars and bars method, the formula for the number of ways to distribute \(n\) identical items into \(k\) distinct categories is given by:
\[ \binom{n+k-1}{k-1} \]
In this scenario:
- \(n = 10\) (total pieces of candy)
- \(k = 4\) (types of candy)
Thus, the calculation becomes:
\[ \binom{10+4-1}{4-1} = \binom{13}{3} \]
Calculating \( \binom{13}{3} \):
\[ \binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286 \]
Therefore, there are 286 different possible mixes of candy. This technique is useful when distributing identical items into distinct categories with no restrictions on how many items each category can receive.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc28801d2-5244-43a8-94e3-88d6acce7d2a%2F2a5551b4-df6b-4c59-a3ac-a728940bca87%2F1xyq1vn_processed.png&w=3840&q=75)
Transcribed Image Text:**Question 8: Combinatorial Candy Mixes**
How many different mixes of candy are possible if a mix consists of 10 pieces of candy and 4 different kinds of candy are available in unlimited quantities?
This problem can be solved using the "stars and bars" theorem in combinatorics. It involves calculating the number of ways to distribute a fixed number of identical items (in this case, 10 candies) into a fixed number of distinct categories (4 types of candy).
**Solution Approach**
Using the stars and bars method, the formula for the number of ways to distribute \(n\) identical items into \(k\) distinct categories is given by:
\[ \binom{n+k-1}{k-1} \]
In this scenario:
- \(n = 10\) (total pieces of candy)
- \(k = 4\) (types of candy)
Thus, the calculation becomes:
\[ \binom{10+4-1}{4-1} = \binom{13}{3} \]
Calculating \( \binom{13}{3} \):
\[ \binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286 \]
Therefore, there are 286 different possible mixes of candy. This technique is useful when distributing identical items into distinct categories with no restrictions on how many items each category can receive.
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