Idea of disproportionation

A species undergoes disproportionation if the same element in it is simultaneously oxidized and reduced.

This is generally possible when the element is in an intermediate oxidation state.

Check each species one by one

(i) $\mathrm{H_2O_2}$

Oxidation state of O is $-1$, which is intermediate between $0$ (in $\mathrm{O_2}$) and $-2$ (in $\mathrm{H_2O}$).

It disproportionates: $2\mathrm{H_2O_2} \rightarrow 2\mathrm{H_2O} + \mathrm{O_2}$

So, yes.

(ii) $\mathrm{ClO_3^-}$

Let oxidation state of Cl be $x$: $x+3(-2)=-1 \Rightarrow x=+5$

Chlorine in $+5$ is intermediate and can undergo disproportionation, e.g. in basic medium: $4\mathrm{ClO_3^-} \rightarrow 3\mathrm{ClO_4^-} + \mathrm{Cl^-}$

So, yes.

(iii) $\mathrm{P_4}$

Phosphorus is in oxidation state $0$.

White phosphorus disproportionates in alkali: $\mathrm{P_4} + 3\mathrm{OH^-} + 3\mathrm{H_2O} \rightarrow \mathrm{PH_3} + 3\mathrm{H_2PO_2^-}$

Here P goes from $0$ to $-3$ and $+1$.

So, yes.

(iv) $\mathrm{Cl_2}$

Chlorine is in oxidation state $0$.

It disproportionates in water/alkali: $\mathrm{Cl_2} + 2\mathrm{OH^-} \rightarrow \mathrm{Cl^-} + \mathrm{ClO^-} + \mathrm{H_2O}$

So, yes.

(v) $\mathrm{Ag}$

Silver is in oxidation state $0$.

It can disproportionate in suitable complexing medium, for example: $2\mathrm{Ag}^+ \rightarrow \mathrm{Ag}^{2+} + \mathrm{Ag}$ but this is for $\mathrm{Ag^+}$, not metallic Ag.

Metallic $\mathrm{Ag}$ itself is not generally considered to undergo disproportionation.

So, no.

(vi) $\mathrm{Cu^{+1}}$

Copper in $+1$ is unstable and disproportionates: $2\mathrm{Cu^+} \rightarrow \mathrm{Cu^{2+}} + \mathrm{Cu}$

So, yes.

(vii) $\mathrm{F_2}$

Fluorine is the most electronegative element. It only gets reduced to $\mathrm{F^-}$ and does not show positive oxidation states in normal chemistry.

Hence it cannot disproportionate.

So, no.

(viii) $\mathrm{NO_2}$

Oxidation state of N is $+4$.

This is intermediate between higher and lower states, and $\mathrm{NO_2}$ disproportionates: $2\mathrm{NO_2} + 2\mathrm{OH^-} \rightarrow \mathrm{NO_2^-} + \mathrm{NO_3^-} + \mathrm{H_2O}$

So, yes.

(ix) $\mathrm{K^+}$

Potassium is already in its maximum stable oxidation state $+1$.

It cannot be oxidized further under normal conditions, so disproportionation is not possible.

So, no.

Count the species that can disproportionate

The species are: $\mathrm{H_2O_2},\ \mathrm{ClO_3^-},\ \mathrm{P_4},\ \mathrm{Cl_2},\ \mathrm{Cu^+},\ \mathrm{NO_2}$

Total number: $6$

Compare with stored answer

Stored correct answer = $6$

Our derived answer = $6$

Hence, the answer agrees.