BIDONGYU

Look at the second column from left.

We know that the carry over is at most $1$ .( See the theorem in 3. )

We deduce that $\textbf{U=0}$ or $\textbf{U=9}$

$\textbf{Case 1:U=0}$

$\begin{array}{cccccc}&&B&I&D&O&N&G\\+&&Y&0&2&0&1&7\\ \hline &&J&I&A&Y&O&0 \end{array}$

We deduce that $\textbf{G=3}$ .

$\begin{array}{cccccc}&&B&I&D&O&N&3\\+&&Y&0&2&0&1&7\\ \hline &&J&I&A&Y&O&0 \end{array}$

We see that there is a carry over at the third column from right.

We deduce that $\textbf{N=8}$ or $\textbf{N=9}$

When $\textbf{N=8}$ ,

$\begin{array}{cccccc}&&B&I&D&O&8&3\\+&&Y&0&2&0&1&7\\ \hline &&J&I&A&Y&O&0 \end{array}$

$\textbf{O=0}$ , which contradicts with $\textbf{U=0}$

$\therefore \textbf{N=9}$

$\begin{array}{cccccc}&&B&I&D&O&9&3\\+&&Y&0&2&0&1&7\\ \hline &&J&I&A&Y&O&0 \end{array}$

$\textbf{O=1}$

$\begin{array}{cccccc}&&B&I&D&1&9&3\\+&&Y&0&2&0&1&7\\ \hline &&J&I&A&Y&1&0 \end{array}$

$\textbf{Y=2}$

$\begin{array}{cccccc}&&B&I&D&1&9&3\\+&&2&0&2&0&1&7\\ \hline &&J&I&A&2&1&0 \end{array}$

The remaining unused digits are $\textbf{4,5,6,7,8}$ .

$\textbf{(B,I,D,J,A)}=\textbf{(4,8,5,6,7)},\textbf{(5,8,4,7,6)},\textbf{(5,4,6,7,8)},\textbf{(6,4,5,8,7)}$

There are a total number of $\textbf{4}$ solutions when $\textbf{U=0}$ .

$\textbf{Case 2: U=9}$

$\begin{array}{cccccc}&&B&I&D&O&N&G\\+&&Y&9&2&0&1&7\\ \hline &&J&I&A&Y&O&9 \end{array}$

We deduce that $\textbf{G=2}$ .

$\begin{array}{cccccc}&&B&I&D&O&N&2\\+&&Y&9&2&0&1&7\\ \hline &&J&I&A&Y&O&9 \end{array}$

Now, let's look at third column, $\textbf{O}$ and $\textbf{Y}$ are different digits so the carry over is $1$ .

$\textbf{N+1}$ $\geq10\Rightarrow$ $\textbf{N}$ $\geq9$

$\textbf{U=9}$ and therefore value of $\textbf{N}$ does not exist.

$\textbf{U=9}$ is unsuitable.

We conclude that there are $\boxed{4}$ solutions to the cryptogram.

You can solve this using code:

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from itertools import permutations
count=0
for b,i,d,o,n,g,y,u,j,a in permutations('0123456789',10):
  if int(b+i+d+o+n+g)+int(y+u+'2017')==int(j+i+a+y+o+u):
    count+=1
print(count)

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Solution:
4