Question

Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f: S `rightarrow` S such that f(m.n) = f(m).f(n) for every m, n ∈ S and m.n ∈ S is equal to ______.

Options

• 460

• 470

• 480

• 490

Answer 1

Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f: S `rightarrow` S such that f(m.n) = f(m).f(n) for every m, n ∈ S and m.n ∈ S is equal to 490.

Explanation:

Given that F(mn) = f(m). f(n)

∵ f(n) = f(1). f(n) `\implies` f(1) = 1

f(4) = f(2.2) = f(2). f(2)`{{:(f(2) = 1 \implies f(4) = 1),(                        or),(f(2) = 2 \implies f(4) = 4):}`

f(6) = f(2) .f(3)`{{:("when"  f(2) = 1 \implies f(6) = f(3)),(f(3) = 1  "to"  7 and f(5)","  f(7) = 1  "to"  7),("So"","  "total" = 7^3  "ways"),(f(2) = 2 \implies f(6) = 2f(3)),(f(3) = 1 or 2 or 3 and f(5)","  f(7) = 1  "to"  7):}`

So total = 3 × 7² ways

Total number of ways = 7³ + 3 × 7²

= 343 + 3 × 49

= 343 + 147

= 490