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Solved Example #7

Saturday, February 9th, 2013

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Question : Find the number of ternary sequences of length 4 where 0 is not followed by 1 and 1 is not followed by 2. (Ternary sequence of length n is a sequence having n terms and each term is either 0,1 or 2)

a. 47

b.37

c. 72

d.54

e.44

 

Answer :

 

Total ternary sequences possible = 3^4 = 81 {There are 4 positions and 3 choices for each position}

 

Invalid sequences:

Sequences which contain 01.

The location for 01 can be selected in 3C2 or 3 ways. {01xx, x01x, xx01}

The other two digits can be selected in 3*3 or 9 ways. {Filling up the x}

No. of invalid sequences = 3*9 = 27

 

Similarly, no. of invalid sequences which contain 12 = 27

 

Some of the invalid sequences have been counted twice. They are:

Type 1 – 0101, 1212, 0112, 1201 – There will be 4 of these

Type 2 – 012x & x120 – There will be 6 of these.

 

Total valid sequences = 27 + 27 – (4 + 6) = 44

 

Total valid sequences = 81 – 44 = 37.Thus, Option B

 

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