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
∴ Total possible functions =1×6=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.
∴ Total functions =2×6=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
∴ Total functions =3×6=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
∴ Total functions =4×6=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
∴ Total possible functions =5×6=30
∴ Total functions from those 5 cases we get =6+12+18+24+30 =90
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