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# Two counting problems

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1. Three friends: Al, Bill, and Carl, are painting their chests with the word CAVALIERS for a basketball game, with at least two letters painted on each friend’s chest. For example, Al might get “CA”, Bill “VAL”, and Carl “IERS”, or Carl could get “CAVA” while Bill gets “LI” and Al gets “ERS”. How many ways can the three friends get their chests painted so that when they stand in the correct order, they spell out the word CAVALIERS?

2. An elaborate code consists of left and right arrows which are placed together to represent words. There are spaces between words. For example, the code of three left and four right arrows arranged with two spaces: >> <> represents a three word code phrase. How many different three-word code phrases in this system use 8 letters?

Jul 15, 2018

#1
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1. Three friends: Al, Bill, and Carl, are painting their chests with the word CAVALIERS for a basketball game, with at least two letters painted on each friend’s chest. For example, Al might get “CA”, Bill “VAL”, and Carl “IERS”, or Carl could get “CAVA” while Bill gets “LI” and Al gets “ERS”. How many ways can the three friends get their chests painted so that when they stand in the correct order, they spell out the word CAVALIERS?

CA   VELIE   RS

Middle

VELIE

VELI

VEL

VE

ELIE

ELI

EL

LIE

LI

IE

So that is 10 ways  if the order the boys stand is is A, Bil Carl

If they can stand in any order then there are   10*3*2 = 60 ways.

Jul 16, 2018
#2
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2. An elaborate code consists of left and right arrows which are placed together to represent words. There are spaces between words. For example, the code of three left and four right arrows arranged with two spaces: >> <> represents a three word code phrase. How many different three-word code phrases in this system use 8 letters?

I do not understand the explanation in question 2. I have no idea what is being described.

Jul 16, 2018
#3
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How many different three-word code phrases in this system use 8 letters?

The posted example is inconsistent with the description preceding it.

This is a valid example: <<  >>>  <

Assumptions: a single bit of “<” or “>” can represent two words. Two bits can represent four words, etc.

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Solution:

Treat as a binary code of 8 bits partitioned into sets of three (3). The value of each partition is an exponent of (2). Multiply these partitions, then multiply by (6) or (3) for the unique permutations of each partition.

8 bits partitioned into sets of (3) where the number of bits is the exponent of (2).

(2^6)*(2^1)*(2^1) =256 | 256*3 =768

(64) words * (2) words * (2) words * (3) unique permutations.

(2^5)*(2^2)*(2^1) = 256 | 256*6 =1536

(32) words * (4) words * (1) word * (6) unique permutations.

(2^3)*(2^3)*(2^2) = 256 | 256*3=768

(2^4)*(2^3)*(2^1) = 256 | 256*6 =1536

(2^4)*(2^2)*(2^2) = 256 | 256*3=768

5376 Total Phrases

GA

Jul 18, 2018