UPSC NDA I 2023 Mathematics Solved Paper

Concept:Use the formula for the number of terms in the expansion of a multinomial.For $(a+b+c)^n$, the number of distinct terms is ${}^{n+3-1}C_{3-1}$.Explanation:First, combine the two factors: $(3x - y)^4(x + 3y)^4 = [(3x - y)(x + 3y)]^4$.Expand the product inside: $(3x - y)(x + 3y) = 3x^2 + 9xy - xy - 3y^2 = 3x^2 + 8xy - 3y^2$.So the expression becomes $(3x^2 + 8xy - 3y^2)^4$.This is of the form $(a+b+c)^n$ with $n=4$ and three distinct terms $a$, $b$, $c$.The number of terms in the expansion is $\;{}^{n+3-1}C_{3-1} = \;{}^{4+2}C_{2} = \;{}^{6}C_{2}$.Compute: $\;{}^{6}C_{2} = \frac{6 \times 5}{2 \times 1} = 15$.Thus, there are 15 terms in the expansion.Answer:15 (Option C).