Discussion

Explanation:

Let the average marks of the three sections A, B and C be a, b and c respectively.
Let the number of students in the three sections A, B and C be n_a, n_b and n_c respectively.

Average marks of sections A and B combined is 56.
⇒ 56 = $\frac{\mathrm{a}\times {\mathrm{n}}_{\mathrm{a}}+\mathrm{b}\times {\mathrm{n}}_{\mathrm{b}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}}$
⇒ an_a + bn_b = 56n_a + 56n_b   ...(1)

Similarly, average marks of sections B and C combined is 60.
⇒ bn_b + cn_c = 60n_b + 60n_c  ...(2)

Similarly, average marks of sections C and A combined is 64.
⇒ cn_c + an_a = 64n_c + 64n_a  ...(3)

Adding (1), (2) and (3), we get
⇒ 2an_a + 2bn_b + 2cn_c = 120n_a + 116n_b + 124n_c
⇒ an_a + bn_b + cn_c = 60n_a + 58n_b + 62n_c

Now, average of all three sections combined = $\frac{\mathrm{a}\times {\mathrm{n}}_{\mathrm{a}}+\mathrm{b}\times {\mathrm{n}}_{\mathrm{b}}+\mathrm{c}\times {\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$

⇒ $\frac{\mathrm{a}\times {\mathrm{n}}_{\mathrm{a}}+\mathrm{b}\times {\mathrm{n}}_{\mathrm{b}}+\mathrm{c}\times {\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$ = $\frac{60{\mathrm{n}}_{\mathrm{a}}+58{\mathrm{n}}_{\mathrm{b}}+62{\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$

⇒ $\frac{\mathrm{a}\times {\mathrm{n}}_{\mathrm{a}}+\mathrm{b}\times {\mathrm{n}}_{\mathrm{b}}+\mathrm{c}\times {\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$ = $\frac{58({\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}})+2{\mathrm{n}}_{\mathrm{a}}+4{\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$

⇒ $\frac{\mathrm{a}\times {\mathrm{n}}_{\mathrm{a}}+\mathrm{b}\times {\mathrm{n}}_{\mathrm{b}}+\mathrm{c}\times {\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$ = $\frac{58({\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}})}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$ + $\frac{2{\mathrm{n}}_{\mathrm{a}}+4{\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$

⇒ $\frac{\mathrm{a}\times {\mathrm{n}}_{\mathrm{a}}+\mathrm{b}\times {\mathrm{n}}_{\mathrm{b}}+\mathrm{c}\times {\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$ = 58 + (positive number)

⇒ $\frac{\mathrm{a}\times {\mathrm{n}}_{\mathrm{a}}+\mathrm{b}\times {\mathrm{n}}_{\mathrm{b}}+\mathrm{c}\times {\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$ > 58

Also, 

⇒ $\frac{\mathrm{a}\times {\mathrm{n}}_{\mathrm{a}}+\mathrm{b}\times {\mathrm{n}}_{\mathrm{b}}+\mathrm{c}\times {\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$ = $\frac{60{\mathrm{n}}_{\mathrm{a}}+58{\mathrm{n}}_{\mathrm{b}}+62{\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$

⇒ $\frac{\mathrm{a}\times {\mathrm{n}}_{\mathrm{a}}+\mathrm{b}\times {\mathrm{n}}_{\mathrm{b}}+\mathrm{c}\times {\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$ = $\frac{62({\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}})-2{\mathrm{n}}_{\mathrm{a}}-4{\mathrm{n}}_{\mathrm{b}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$

⇒ $\frac{\mathrm{a}\times {\mathrm{n}}_{\mathrm{a}}+\mathrm{b}\times {\mathrm{n}}_{\mathrm{b}}+\mathrm{c}\times {\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$ = $\frac{62({\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}})}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$ - $\frac{2{\mathrm{n}}_{\mathrm{a}}+4{\mathrm{n}}_{\mathrm{b}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$

⇒ $\frac{\mathrm{a}\times {\mathrm{n}}_{\mathrm{a}}+\mathrm{b}\times {\mathrm{n}}_{\mathrm{b}}+\mathrm{c}\times {\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$ = 62 - (positive number)

⇒ $\frac{\mathrm{a}\times {\mathrm{n}}_{\mathrm{a}}+\mathrm{b}\times {\mathrm{n}}_{\mathrm{b}}+\mathrm{c}\times {\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$ < 62

∴ 58 < $\frac{\mathrm{a}\times {\mathrm{n}}_{\mathrm{a}}+\mathrm{b}\times {\mathrm{n}}_{\mathrm{b}}+\mathrm{c}\times {\mathrm{n}}_{\mathrm{c}}}{{\mathrm{n}}_{\mathrm{a}}+{\mathrm{n}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{c}}}$ < 62

Hence, option (c)

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