A mistake often made by first year calculus students is in thinking that
(a+b)n = an + bn. This mistake has acquired the nickname of the “Freshman’s Dream”.
A related result (that is true) known as the the “Freshman’s Dream Theorem” is that
(a+b)n = an + bn (mod n) when n is a prime or a Carmichael number.
Source: Pratibha Ghatage and Brian Scott, When is (a+b)n = an + bn (mod n)?, College Mathematics Journal, Vol. 36, No. 4 (Sep 2005), p. 322.
If there was a calculus “Professor’s Dream” it would likely be that all students carefully read (and then read again several more times) the following pages before they reach calculus:
Paul's Online Math Notes: Common Math Errors
The Most Common Errors in Undergraduate Mathematics