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Hello there's a question where you need to find the total possible injective and surjective possibilities. Could someone verify my answer? I'm almost certain of answer a, but want to know if b is also correct?

If you know if its true can you reply, it would be very helpful. If it's wrong could you explain why?

Thanks in advance

question

assume |X|=2 and |Y|=2016

a) How many injective functions are there from X to Y? My answer = $\frac{2016!}{2014!}$

b) How many surjective functions are there from Y to X? My answer = $2^{2016}-2$

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  • $\begingroup$ Whether or not these answers are right, you don't seem to have learned much mathematics. Solving a problem is more than "finding a formula" and asking whether it's right. If you edit the question to tell us why you think these formulas count what you are interested in, perhaps we can help. Note: that quotient of factorials simplifies a lot. $\endgroup$ Jan 9 at 16:39
  • $\begingroup$ This tutorial explains how to typeset mathematics on this site. $\endgroup$ Jan 9 at 16:44
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Your answer for injections is correct.

Your answer for surjections is incorrect. In total we have $2^{2016}$ functions. However the only way a function is non-surjective in this case is if both of the elements in $X$ are mapped to the same element in $Y$, and there are $2016$ elements in $Y$.

So there are actually $2^{2016}-2016$ injections.

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  • $\begingroup$ A wow indeed, thank you so much :) $\endgroup$
    – TheCreator
    Jan 9 at 16:53
  • $\begingroup$ You are confused somehow, the correct answer is $2^{2016}-2$. You need to subtract the two constant maps. $\endgroup$ Jan 9 at 17:26
  • $\begingroup$ You talk about the elements of X being mapped to elements of Y, but it should be the other way around. $\endgroup$ Jan 9 at 17:28

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