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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Permutation and Combination

Olympiad workbook · IOQM · Permutation and Combination

Given a 2 × 2 tile and seven dominoes ( 2 × 1 tile), find the...

Given a 2×22 \times 2 tile and seven dominoes ( 2×12 \times 1 tile), find the number of ways of tiling (that is, cover without leaving gaps and without overlapping of any two tiles) a 2×72 \times 7 rectangle using some of these tiles.Given a 2 × 2 tile and seven dominoes ( 2 × 1 tile), find the number of ways of tiling (that is, cover without leaving

Correct answer

85

Step-by-step solution

Given a 2 × 2 tile and seven dominoes ( 2 × 1 tile), find the number of ways of tiling (that is, cover without leaving
an=2an−2+an−1a1=1,a2=3a3=2a1+a2=2(1)+3=5a4=2a2+a3=2(3)+5=11a5=2a3+a4=2(5)+11=21a6=2(11)+21=43a7=2(21)+43=42+43=85\begin{aligned} & a_n=2 a_{n-2}+a_{n-1} \\ & a_1=1, a_2=3 \\ & a_3=2 a_1+a_2=2(1)+3=5 \\ & a_4=2 a_2+a_3=2(3)+5=11 \\ & a_5=2 a_3+a_4=2(5)+11=21 \\ & a_6=2(11)+21=43 \\ & a_7=2(21)+43=42+43=85\end{aligned}
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