To organize the given expressions from greatest to least by the number of terms, we need to determine the number of terms present in each expression. Let us analyze each expression one by one:
1. Expression [tex]\( I \)[/tex]: [tex]\( x + 2xy z \)[/tex]
- This has 2 terms: [tex]\( x \)[/tex] and [tex]\( 2xy z \)[/tex].
2. Expression [tex]\( II \)[/tex]: [tex]\( 3x + y + z \)[/tex]
- This has 3 terms: [tex]\( 3x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex].
3. Expression [tex]\( III \)[/tex]: [tex]\( 2x^2 y + x^2 - 3x + 4 \)[/tex]
- This has 4 terms: [tex]\( 2x^2 y \)[/tex], [tex]\( x^2 \)[/tex], [tex]\( -3x \)[/tex], and [tex]\( 4 \)[/tex].
4. Expression [tex]\( IV \)[/tex]: [tex]\( 9x^2 yz \)[/tex]
- This has 1 term: [tex]\( 9x^2 yz \)[/tex].
Next, we organize the expressions based on the number of terms from greatest to least:
- Expression [tex]\( III \)[/tex] has 4 terms.
- Expression [tex]\( II \)[/tex] has 3 terms.
- Expression [tex]\( I \)[/tex] has 2 terms.
- Expression [tex]\( IV \)[/tex] has 1 term.
Arranging these in order, we get:
[tex]\[
III, II, I, IV
\][/tex]
Therefore, the correct order from greatest to least by number of terms is:
[tex]\[
\boxed{III, II, I, IV}
\][/tex]
