The total number of functions, [ f:1,2,3,4 1,2,3,4,5,6 ] such that f(1)+f(2)=f(3), is equa…

Question

The total number of functions,

$$
f:\{1,2,3,4\} \rightarrow\{1,2,3,4,5,6\}
$$
 such that $f(1)+f(2)=f(3)$, is equal to :

Accepted Answer

Given, $f(1) + f(2) = f(3)$

It means $f(1),f(2)$ and $f(3)$ are dependent on each other. But there is no condition on $f(4)$, so $f(4)$ can be $f(4) = 1,2,3,4,5,6$.

For $f(1),f(2)$ and we have to find how many functions possible which will satisfy the condition $f(1) + f(2) = f(3)$

Case 1 :

When $f(3) = 2$ then possible values of $f(1)$ and $f(2)$ which satisfy $f(1) + f(2) = f(3)$ is $f(1) = 1$ and $f(2) = 1$.

And $f(4)$ can be = 1, 2, 3, 4, 5, 6

$$\therefore$$ Total possible functions $$=1\times6=6$$

Case 2 :

When $f(3) = 3$ then possible values

(1) $f(1) = 1$ and $f(2) = 2$

(2) $f(1) = 2$ and $f(2) = 1$

And $f(4)$ can be = 1, 2, 3, 4, 5, 6.

$$\therefore$$ Total functions $$ = 2 \times 6 = 12$$

Case 3 :

When $f(3) = 4$ then

(1) $f(1) = 1$ and $f(2) = 3$

(2) $f(1) = 2$ and $f(2) = 2$

(3) $f(1) = 3$ and $f(2) = 1$

And $f(4)$ can be = 1, 2, 3, 4, 5, 6

$$\therefore$$ Total functions $$ = 3 \times 6 = 18$$

Case 4 :

When $f(3) = 5$ then

(1) $f(1) = 1$ and $f(4) = 4$

(2) $f(1) = 2$ and $f(4) = 3$

(3) $f(1) = 3$ and $f(4) = 2$

(4) $f(1) = 4$ and $f(4) = 1$

And $f(4)$ can be  = 1, 2, 3, 4, 5 and 6

$$\therefore$$ Total functions $$ = 4 \times 6 = 24$$

Case 5 :

When $f(3) = 6$ then

(1) $f(1) = 1$ and $f(2) = 5$

(2) $f(1) = 2$ and $f(2) = 4$

(3) $f(1) = 3$ and $f(2) = 3$

(4) $f(1) = 4$ and $f(2) = 2$

(5) $f(1) = 5$ and $f(2) = 1$

And $f(4)$ can be = 1, 2, 3, 4, 5 and 6

$$\therefore$$ Total possible functions $$ = 5 \times 6 = 30$$

$$\therefore$$ Total functions from those 5 cases we get

$= 6 + 12 + 18 + 24 + 30$

$= 90$

Solutions · 1