Olympiad workbook
- Exam
- Olympiad workbook
- Subject
- IOQM
- Chapter
- Permutation and Combination
Olympiad workbook · IOQM · Permutation and Combination
A 1 × n rectangle (n ≥ 1) is divided into n unit (1 × 1) squares....
A rectangle is divided into unit squares. Each square of this rectangle is coloured red, blue or green. Let be the number of colourings of the rectangle in which there are an even number of red squares. What is the largest prime factor of ? (The number of red squares can be zero)
Correct answer
37
Step-by-step solution
f(n)={ }^n C_0 2^n+{ }^n C_2 2^{n-2}+{ }^n C_4 2^{n-4}+\ldots \ldots \ldots \ldots .(2+1)^{\mathrm{n}}={ }^{\mathrm{n}} \mathrm{C}_0 2^{\mathrm{n}}+{ }^{\mathrm{n}} \mathrm{C}_1 2^{\mathrm{n}-1}+{ }^{\mathrm{n}} \mathrm{C}_2 2^{\mathrm{n}-2}+{ }^{\mathrm{n}} \mathrm{C}_3 2^{\mathrm{n}-3}+(2-1)^n={ }^n C_0 2^n-{ }^n C_1 2^{n-1}+{ }^n C_2 2^{n-2}-{ }^n C_3 2^{n-3}+....(ii)3^n+1=2
Largest prime factor is 37 .