Based on a criterion of permutation polynomials of the form x rf(x q-1/m) by Wan and Lidl (1991) and some very elementary techniques we show existence of permutation binomials of the following forms (i) x(x 2n-1/3 + a) ∈ double-struck F 2n[x], for n > 4 (ii) x 22n-1/2n-1+1 + ax = x 2n+2 + ax ∈ double-struck F 22n[x], for n ≥ 3. In (i), we extend a result of Carlitz (1962) for even characteristic. Moreover we present the count of such permutation binomials when a is in a certain subfield of double-struck F 2n. In (ii), we reprove, using much simpler technique, a recent result of Charpin and Kyureghyan (2008) and give the number of permutation binomials of this form. Finally, we discuss some cryptographic relevance of these results. © 2012 Springer-Verlag.