To calculate the number of ways 12 hens can nest in 5 nesting boxes, we can use the formula for distributing \( n \) indistinguishable items (hens) into \( r \) distinguishable boxes (nesting boxes). This is a combinatorial problem known as the "stars and bars" theorem.
The formula for the number of ways to distribute \( n \) indistinguishable objects into \( r \) distinguishable boxes is given by:
\[ \binom{n + r - 1}{r - 1} \]
In this case, \( n = 12 \) (the hens) and \( r = 5 \) (the nesting boxes). Plugging in these values:
\[ \binom{12 + 5 - 1}{5 - 1} = \binom{16}{4} \]
Now, we calculate \( \binom{16}{4} \):
\[ \binom{16}{4} = \frac{16!}{4!(16-4)!} = \frac{16!}{4! \cdot 12!} \]
Calculating the factorials, we get:
\[ \binom{16}{4} = \frac{16 \times 15 \times 14 \times 13}{4 \times 3 \times 2 \times 1} \]
Now, calculating the numerator:
\[ 16 \times 15 = 240 \] \[ 240 \times 14 = 3360 \] \[ 3360 \times 13 = 43680 \]
Calculating the denominator:
\[ 4 \times 3 \times 2 \times 1 = 24 \]
Finally, dividing the results:
\[ \binom{16}{4} = \frac{43680}{24} = 1820 \]
Thus, the number of ways the 12 hens can nest in the 5 nesting boxes is \( \boxed{1820} \).