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Chapter 11: Problem 55

Find the number of different signals consisting of eight flags that can be made using three white flags, four red flags, and one blue flag.

Short Answer

Expert verified
The total different signals that can be made using three white flags, four red flags, and one blue flag is 280.

Step by step solution

01

Understanding permutations with repetition

Firstly, it's necessary to understand the basic principle of permutations with repetition. When we deal with permutations, we consider the order of the objects. But when the objects are repeated among themselves, moving them around doesn't change anything as they are identical. Thus, we need to adjust our approach in calculating permutations to account for these repetitions.
02

Calculate total permutations

In total, we have eight flags which can be arranged in 8! (factorial 8) ways. The factorial function (denoted with an exclamation mark) means that we are multiplying all positive whole numbers up to that number. In this case, \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320\).
03

Calculate repetitions

Now, we have to calculate how many of these arrangements are just repetitions. We have three white flags, which among themselves can be arranged in \(3! = 3 \times 2 \times 1 = 6\) ways. Similarly, we have four red flags, which can be arranged in \(4! = 4 \times 3 \times 2 \times 1 = 24\) ways. The blue flag is unique, so it can be arranged in only one way.
04

Apply formula for permutations with repetition

To find the number of unique signals, we divide the total permutations by the repeating permutations. The formula for permutations with repetition is \(\frac{n!}{r1! \times r2! \times ... \times rk!}\) where 'n' is the total number of objects and 'r1', 'r2'... 'rk' are the number of each type of repeating objects. In our scenario, we substitute n with 8, r1 with 3 and r2 with 4. Thus, the formula will be \(\frac{8!}{3! \times 4!} = \frac{40320}{6 \times 24} = 280\).

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Most popular questions from this chapter

Involve computing expected values in games of chance. For many years, organized crime ran a numbers game that is now run legally by many state governments. The player selects a three-digit number from 000 to 999 . There are 1000 such numbers. A bet of \(\$ 1\) is placed on a number, say number 115. If the number is selected, the player wins \(\$ 500\). If any other number is selected, the player wins nothing. Find the expected value for this game and describe what this means.

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