This question asks to identify the correct statements regarding nucleophilic substitution reactions for four given compounds.

Let's first analyze the structure of each compound and its tendency to undergo ${S_N}1$ or ${S_N}2$ reactions.

Compound I: $CH_3-CH_2-Br$ (Ethyl bromide)

• This is a primary ($1^\circ$) alkyl halide.
• It is sterically unhindered, favoring backside attack by a nucleophile.
• The primary carbocation ($CH_3-CH_2^+$) that would be formed in an ${S_N}1$ reaction is highly unstable.
• Therefore, compound I strongly favors the ${S_N}2$ mechanism.

Compound II: $H_2C=CH-CH_2-Br$ (Allyl bromide)

• This is a primary ($1^\circ$) allylic halide.
• Being a primary halide, it is unhindered and thus a good substrate for ${S_N}2$ reactions. The rate is even enhanced due to the stabilization of the transition state by the adjacent $\pi$-bond.
• The allyl carbocation ($H_2C=CH-CH_2^+$) formed in an ${S_N}1$ reaction is stabilized by resonance, so it can also undergo ${S_N}1$ reactions under appropriate conditions (polar protic solvent, weak nucleophile).

Compound III: $(CH_3)_3C-Br$ (tert-Butyl bromide)

• This is a tertiary ($3^\circ$) alkyl halide.
• It is sterically hindered, preventing backside attack required for the ${S_N}2$ mechanism.
• The tertiary carbocation ($(CH_3)_3C^+$) formed upon departure of the bromide ion is very stable due to hyperconjugation and inductive effects.
• Therefore, compound III strongly favors the ${S_N}1$ mechanism.

Compound IV: $C_6H_5-CH(CH_3)-Br$ (1-Bromo-1-phenylethane)

• This is a secondary ($2^\circ$) benzylic halide.
• The carbocation formed in an ${S_N}1$ reaction, $C_6H_5-C^+H(CH_3)$, is a secondary benzylic carbocation. It is highly stabilized by resonance with the benzene ring.
• This high stability of the carbocation strongly favors the ${S_N}1$ mechanism.
• However, being a secondary halide, it can also undergo ${S_N}2$ reactions, especially with strong nucleophiles in polar aprotic solvents.

Now, let's evaluate each statement:

A: I and III follow ${S_N}1$ mechanism

• Compound I ($CH_3-CH_2-Br$) is a primary halide and undergoes ${S_N}2$ reactions, not ${S_N}1$.
• Compound III ($(CH_3)_3C-Br$) is a tertiary halide and undergoes ${S_N}1$ reactions.
• Since the statement is not true for compound I, statement A is incorrect.

B: I and II follow ${S_N}2$ mechanism

• Compound I ($CH_3-CH_2-Br$) is a classic example of a substrate that undergoes ${S_N}2$ reactions.
• Compound II ($H_2C=CH-CH_2-Br$) is a primary allylic halide, which is unhindered and readily undergoes ${S_N}2$ reactions.
• Thus, statement B is correct.

C: Compound IV undergoes inversion of configuration

• Inversion of configuration is the stereochemical outcome of an ${S_N}2$ reaction.
• Compound IV ($C_6H_5-CH(CH_3)-Br$) is a chiral secondary benzylic halide. While it prefers the ${S_N}1$ pathway in many conditions due to its stable carbocation, it can undergo an ${S_N}2$ reaction under conditions that favor it (e.g., a strong, non-bulky nucleophile in a polar aprotic solvent).
• Since an ${S_N}2$ pathway is possible for compound IV, it can undergo inversion of configuration.
• Therefore, statement C is correct.

D: The order of reactivity for I, III and IV is : $IV > I > III$

• The order of reactivity depends on the reaction mechanism.
• For ${S_N}1$ reactions (favored by III and IV), reactivity depends on carbocation stability: $IV > III >> I$.
• For ${S_N}2$ reactions (favored by I), reactivity depends on steric hindrance: $I > IV > III$ (III is essentially unreactive).
• The given order $IV > I > III$ is inconsistent. The part $I > III$ suggests an ${S_N}2$ trend, while $IV > I$ suggests an ${S_N}1$ trend. If we compare under conditions that allow both (like solvolysis), the order would be determined by carbocation stability, i.e., $IV > III > I$. The given order does not match any standard reactivity series.
• Therefore, statement D is incorrect.

Based on the analysis, the correct statements are B and C.

Comparison with Stored Answer: The stored answer is A, B, C. This is logically inconsistent. Statement A claims compound I follows the ${S_N}1$ mechanism, while statement B claims it follows the ${S_N}2$ mechanism. Both statements cannot be simultaneously correct. As established, compound I follows the ${S_N}2$ mechanism, making statement A incorrect and statement B correct. Therefore, the stored answer key appears to be erroneous.