Which of the following combinations is the most likely outcome of the experiment ?

here most likely means that we have to pick the options which has the highest value so we can compare the options and select the appropriate one.

$A = P(3 {\color{Green} G},4{\color{Red} R}) = {}^7C_3. {\color{Green}{\left(\frac{4}{6}\right)^3}}{\color{Red}{\left(\frac{2}{6}\right)^4}} $

$B = P(4 {\color{Green} G},3{\color{Red} R}) = {}^7C_4. {\color{Green}{\left(\frac{4}{6}\right)^4}}{\color{Red}{\left(\frac{2}{6}\right)^3}} $

$C = P(5 {\color{Green} G},2{\color{Red} R}) = {}^7C_5. {\color{Green} {\left(\frac{4}{6}\right)^5}}{\color{Red} {\left(\frac{2}{6}\right)^2}} $

$D = P(6 {\color{Green} G},1{\color{Red} R}) = {}^7C_6.{\color{Green} { \left(\frac{4}{6}\right)^6}}{\color{Red}{\left(\frac{2}{6}\right)^1}}$

since we are comparing so multiplying each option with $6^{7}$ will not change the result.

$A = {}^7C_3. {\color{Green}{\left(\frac{4}{1}\right)^3}}{\color{Red}{\left(\frac{2}{1}\right)^4}} $

$B = {}^7C_4. {\color{Green}{\left(\frac{4}{1}\right)^4}}{\color{Red}{\left(\frac{2}{1}\right)^3}} $

$C = {}^7C_5. {\color{Green} {\left(\frac{4}{1}\right)^5}}{\color{Red} {\left(\frac{2}{1}\right)^2}} $

$D = {}^7C_6.{\color{Green} { \left(\frac{4}{1}\right)^6}}{\color{Red}{\left(\frac{2}{1}\right)^1}} $

Now convert each of the ${\color{Green} {4^{x}}}$ term into ${\color{Green} {2^{2x}}}$ term and multiply them with ${\color{Red} {2^{y}}}$

$A = {}^7C_3. 2^{10}$

$B = {}^7C_4. 2^{11}$

$C = {}^7C_5. 2^{12}$

$D = {}^7C_6. 2^{13}$

since we are comparing so dividing each option with $2^{10}$ will not change the result.

$A = {}^7C_3.=35$

$B = {}^7C_4. 2=70$

$C = {}^7C_5. 2^{2}=84$

$D = {}^7C_6. 2^{3}=56$

$\because$ Option $C$ has the highest value so it is the correct answer.

$\therefore$ option $C$ is the correct answer.