A simple expansion problem

When we expand this expression, we have:

$a^2+b^2+\ldots+i^2$ , which has $9$ terms

$ab+ac+\ldots+ai$ , which has $8$ terms

$bc+bd + \ldots + bi$ , which has $7$ terms (we don't add $ba$ because it's the same as $ab$ )

$cd+ce+ \ldots + ci$ , which has $6$ terms

$de + df + \ldots + di$ , which has $5$ terms

$ef + eg +eh +ei$ , which has $4$ terms

$fg+fh+fi$ , which has $3$ terms

$gh+gi$ , which has $2$ terms, and

$hi$ , which is the final term in the expression

Total number of terms $= 9 +8 +7 +6+5+4+3+2+1 = \boxed{45}$

(Combinations of 2 out of 9) plus( 9) equals 45 terms combinations of 2 out of 9 for heterogeneous products and 9 for homogeneous products.