8) In a six-cylinder V6 engine, the even-numbered cylinders are on the left and

Question

8) In a six-cylinder V6 engine, the even-numbered cylinders are on the left and the odd numbered cylinders are on the right. A good firing order is a sequence of the numbers 1 through 6 in which the sides alternate. a) How many good firing sequences are there that start on the left? b) How many possible good firing sequences are there? c) How many possible good firing sequences are there for an eight cylinder V8 engine? Answer a) Answer b) Answer c) Big Idea Assignment: Combinatorics MATH 135, Fall 2017 3

Explanation / Answer

Left : 2,4,6

Right :1,3,5

(a)

if it starts on the left we have following number of option for each position in the sequence

1st - 3

2nd - 3

3nd - 2

4th - 2

5th - 1

6th - 1

total = 3*3*2*2*1*1 = 36

(b)

it can start with any number

1st - 6

2nd - 3

3nd - 2

4th - 2

5th - 1

6th - 1

total = 6*3*2*2*1*1 = 72

alternate way to do is, there are only 2 options here. either to start from left or from right.

either side is same and have 36 ways, so total are 72

(c)

Similarly as above

total = 2*4*4*3*3*2*2*1*1 = 1152

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