How many sequences of 0s and 1s of length 19 are there that begin with a 0, end with a 0, contain no two consecutive 0s, and con

Question

2 years ago

10

tain no three consecutive 1s

1 answer:

sladkih [1.3K] · Answer 1 · 2023-03-30T20:45:17+03:00

65 sequences.

Lets solve the problem,

The last term is 0.

To form the first 18 terms, we must combine the following two sequences:

0-1 and 0-1-1

Any combination of these two sequences will yield a valid case in which no two 0's and no three 1's are adjacent

So we will combine identical 2-term sequences with identical 3-term sequences to yield a total of 18 terms, we get:

2x + 3y = 18

Case 1: x=9 and y=0

Number of ways to arrange 9 identical 2-term sequences = 1

Case 2: x=6 and y=2

Number of ways to arrange 6 identical 2-term sequences and 2 identical 3-term sequences =8!6!2!=28=8!6!2!=28

Case 3: x=3 and y=4

Number of ways to arrange 3 identical 2-term sequences and 4 identical 3-term sequences =7!3!4!=35=7!3!4!=35

Case 4: x=0 and y=6

Number of ways to arrange 6 identical 3-term sequences = 1

Total ways = Case 1 + Case 2 + Case 3 + Case 4 = 1 + 28 + 35 + 1 = 65

Hence the number of sequences are 65.

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