Total number of increasing functions
How many strictly increasing functions f are possible such that f : A → B , where A = { a 1 , a 2 , a 3 , a 4 , a 5 , a 6 } , and B = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } and a i + 1 > a i ∀ i ∈ N and f ( a i ) = i ?
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The answer is 28.
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1 solution
How could you conclude that f ( a ) = 1 . In the question, it's not given that f ( a i ) = i .
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I also had same doubt
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Thanks. I have edited that f ( a i ) = i . It used to be a i = i .
Indeed ,, whats more beautiful is that only f(a1) not = 1 is needed along with the general increasing condition, the other possibilities of f(ai)=i is automatically blocked
Can't the value of f ( a ) be same for 2 different values of a ? The function is increasing not strictly increasing.
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Thanks. I have added that the function is strictly increasing.
Over rated...
How ??? f(ai) = i. Is a increasing function... It's it?? Should answer be 9 choose 6. 9C6.
It is impossible that f ( a ) = 1 , for any element in A . Hence we need to only look to connect all the elements other than 1 . We now have ( 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 ) . It can be observed that choosing a combination of any six numbers from this set will give us one and exactly one f such that the conditions are satisfied. (make f by choosing any 6 numbers and connecting them in ascending order from a 1 to a 6 )
Hence the answer is 8 c h o o s e 6 = 2 8