JEE MAIN - Mathematics (2021 - 27th July Morning Shift - No. 23)
Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f : S $$\to$$ S
such that f(m . n) = f(m) . f(n) for every m, n $$\in$$ S and m . n $$\in$$ S is equal to _____________.
such that f(m . n) = f(m) . f(n) for every m, n $$\in$$ S and m . n $$\in$$ S is equal to _____________.
Answer
490
Explanation
F(mn) = f(m) . f(n)
Put m = 1 f(n) = f(1) . f(n) $$\Rightarrow$$ f(1) = 1
Put m = n = 2
$$f(4) = f(2).f(2)\left\{ \matrix{ f(2) = 1 \Rightarrow f(4) = 1 \hfill \cr or \hfill \cr f(2) = 2 \Rightarrow f(4) = 4 \hfill \cr} \right.$$
Put m = 2, n = 3
$$f(6) = f(2).f(3)\left\{ \matrix{ when\,f(2) = 1 \hfill \cr f(3) = 1\,to\,7 \hfill \cr \hfill \cr f(2) = 2 \hfill \cr f(3) = 1\,or\,2\,or\,3 \hfill \cr} \right.$$
f(5), f(7) can take any value
Total = (1 $$\times$$ 1 $$\times$$ 7 $$\times$$ 1 $$\times$$ 7 $$\times$$ 1 $$\times$$ 7) + (1 $$\times$$ 1 $$\times$$ 3 $$\times$$ 1 $$\times$$ 7 $$\times$$ 1 $$\times$$ 7)
= 490
Put m = 1 f(n) = f(1) . f(n) $$\Rightarrow$$ f(1) = 1
Put m = n = 2
$$f(4) = f(2).f(2)\left\{ \matrix{ f(2) = 1 \Rightarrow f(4) = 1 \hfill \cr or \hfill \cr f(2) = 2 \Rightarrow f(4) = 4 \hfill \cr} \right.$$
Put m = 2, n = 3
$$f(6) = f(2).f(3)\left\{ \matrix{ when\,f(2) = 1 \hfill \cr f(3) = 1\,to\,7 \hfill \cr \hfill \cr f(2) = 2 \hfill \cr f(3) = 1\,or\,2\,or\,3 \hfill \cr} \right.$$
f(5), f(7) can take any value
Total = (1 $$\times$$ 1 $$\times$$ 7 $$\times$$ 1 $$\times$$ 7 $$\times$$ 1 $$\times$$ 7) + (1 $$\times$$ 1 $$\times$$ 3 $$\times$$ 1 $$\times$$ 7 $$\times$$ 1 $$\times$$ 7)
= 490
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