In the given statement, the expression becomes a whole number only when the powers of all the prime numbers are also whole numbers.  Let us first simplify the expression a bit by expressing all terms in terms of prime numbers.   $$sqrt[3]{7^a imes 35^{b+1} imes 20^{c+2}}$$ $$Rightarrow sqrt[3]{7^a imes 5^{b+1} imes 7^{b+1} imes 2^{2(c+2)} imes 5^{c+2}}$$ $$Rightarrow sqrt[3]{2^{2c+4} imes 5^{b+c+3}  imes 7^{a+b+1}}$$
$$Rightarrow 2^{frac{2c+4}{3}} 5^{frac{b+c+3}{3}} 7^{frac{a+b+1}{3}} $$
Now, from the given options, we can put in values of the variables and check the exponents of all the numbers.  Option A : a = 2, b = 1, c = 1 :  In this case, we can see that exponent of 5 ie $$frac{b+c+3}{3} = frac{5}{3} $$ is not a whole number.  Option B : a = 1, b = 2, c = 2 In this case, we can see that exponent of 2 ie $$frac{2c+4}{3} = frac{8}{3} $$ is not a whole number.  Option C : a = 2, b = 1, c = 2 In this case, we can see that exponent of 2 ie $$frac{2c+4}{3} = frac{8}{3} $$ is not a whole number.  Option D : a = 3, b = 1, c = 1 In this case, we can see that exponent of 5 ie $$frac{b+c+3}{3} = frac{5}{3} $$ is not a whole number.  Option E : a = 3, b = 2, c = 1 In this case, we can see that all exponents are whole numbers.  Thus, option E is the correct option.