Question 1162652: In a restaurant, 10 customers are lined up to order. The first customer in line is assigned the number 1, the second customer in line is assigned the number 2, and so on, with the tenth customer being assigned the number 10. Each customer will pay for their order in exactly one of three different ways (cash, credit or debit) according to the following guidelines:
• Customers assigned the numbers p and q use different methods to pay whenever p − q is equal to an odd number.
• It is not required that all 3 payment methods are used.
In total, how many different ways can the 10 customers pay?
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
I can draw this out on paper; but I don't know how successful it will be in a typed response....
Call the three methods of paying A, B, and C.
The first customer can choose any of the 3 methods of payment.
The second customer has 2 choices, because it must be different that customer 2.
So there are 3*2=6 possible sequences of payment methods for the first two customers.
We will come back to that at the end of the analysis.
For the rest of the analysis, we will assume the first two customers have picked payment methods:
AB
(3) customer #3:
The third customer can choose any method not used by customer #2:
AB --> ABA, ABC
So 2 possible sequences for customers 1-3.
(4) customer #4:
This customer's choice must be different than the choices of both #1 and #3.
ABA --> ABAB, ABAC
ABC --> ABCB
So 3 possible sequences for customers 1-4.
(5) customer #5:
This customer's choice must be different than the choices of both #2 and #4.
ABAB --> ABABA, ABABC
ABAC --> ABACA
ABCB --> ABCBA, ABCBC
So 5 possible sequences for customers 1-5.
(6) customer #6:
This customer's choice must be different than the choices of both #3 and #5.
ABABA --> ABABAB, ABABAC
ABABC --> ABABCB
ABACA --> ABACAB, ABACAC
ABCBA --> ABCBAB
ABCBC --> ABCBCA, ABCBCB
So 8 possible sequences for customers 1-6.
I'll go one more step with the detailed analysis....
(6) customer #7:
This customer's choice must be different than the choices of both #4 and #6.
ABABAB --> ABABABA, ABABABC
ABABAC --> ABABACA
ABABCB --> ABABCBA, ABABCBC
ABACAB --> ABACABA
ABACAC --> ABACACA, ABACACB
ABCBAB --> ABCBABA, ABCBABC
ABCBCA --> ABCBCAC
ABCBCB --> ABCBCBA,ABCBCBC
So 13 possible sequences for customers 1-7.
It's hard to see in this typed response, but I can see it on my diagram with pencil and paper: We have a Fibonacci sequence.
Continuing the Fibonacci pattern, we have, for the arbitrary choice AB for the first two customers, 55 different possible sequences for the 10 customers.
And finally, since there are 3*2=6 different ways for the first two customers to choose their payment methods, the total number of ways the 10 customers can pay is 6*55 = 330.
ANSWER: 330 different ways
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