Error check in book - Jensen inequality and binomial coefficient

Question

I'm reading "Ramsey Theory" by Graham, Rothschild and Spencer, 2nd edition (see here). Page 111 & 112, they state

1. For a given $a$, the function $f:x\mapsto \binom{x}{a}$ is concave
2. For a given concave function, with $\bar{x}=(x_1+\ldots+x_n)/n$ $$ \sum_{i=1}^n f(x_i) \geq nf(\bar{x})$$
3. So that we have $$\sum_i \binom{x_i}{a} \geq n \binom{\frac{1}{n}\sum_i x_i}{a}$$

The second point is, without stating it, Jensen inequality. But if I'm not mistaken, Points 1 and 2 are backward. The binomial function is convex, and the Jensen inequality is stated at reverse (this is the statement for convex functions).

In the end, these two mistakes cancel, giving a good results.

Could anyone confirm?