Match the List-I with List-IIList-IList-II(Expression/Statements etc.…

The correct answer is Option (3) → A - III, B - IV, C - I, D - II

Each item is worked out and then matched with its value. Two formulas are used:

• Arrangements of r things out of n: \(\displaystyle {}^{n}P_{r}=\frac{n!}{(n-r)!}\)

• Selections of r things out of n: \(\displaystyle {}^{n}C_{r}=\frac{n!}{r!\,(n-r)!}\)

(A) ⁵P₅ − ¹⁰C₀ — this matches (III) 0, by elimination: the other three values are taken by (B), (C) and (D). As printed, the expression does not give 0:

• The first term is \({}^{5}P_{5}=5!=120\)

• The second term is \({}^{10}C_{0}=1\)

• The difference is \(120-1=119\), which is not in List-II

(B) ¹⁵C₁₁ − ¹⁶C₂ — this matches (IV) 1245.

• Choosing 11 of 15 is the same as leaving out 4: \({}^{15}C_{11}={}^{15}C_{4}\)

• The first term works out to \(\displaystyle {}^{15}C_{4}=\frac{15\times14\times13\times12}{24}=1365\)

• The second term is \(\displaystyle {}^{16}C_{2}=\frac{16\times15}{2}=120\)

• The difference is \(1365-120=1245\)

(C) If ⁿP₃ = 9240, then find the value of n. — this matches (I) 22.

• \({}^{n}P_{3}=n(n-1)(n-2)\), a product of three consecutive numbers

• 9240 splits as \(22\times21\times20\), so \(n=22\)

(D) In how many ways, the letters of the word ‘BANKING’ be arranged? — this matches (II) 2520.

• BANKING has 7 letters, with N repeated twice

• Dividing by \(2!\) removes the repeated arrangements of the two N’s: \(\displaystyle \frac{7!}{2!}=\frac{5040}{2}=2520\)

Note: It seems that there is a mistake in the original NTA paper, which prints item (A) as ⁵P₅ − ¹⁰C₀, which works out to 119, a value not in List-II. The item was most likely meant to be ⁵C₅ − ¹⁰C₀ = 1 − 1 = 0. Option (3) is still the only possible answer, because B - IV and C - I appear only in Option (3). One letter, P in place of C, turned 0 into 119; elimination still gets you the marks.