JAMIA 2026 — Mathematics PYQ

1. Analyze the Letters of the Given Word:

The word "BANANA" consists of 6 letters:

• B $\to 1$

• A $\to 3$

• N $\to 2$

We need to form words of length 3. Since there are repeated letters available, we must break this problem down into distinct scenarios based on the types of letters selected.

Case 1: All 3 letters are identical

The only letter available 3 times is A.

• Selection: $\{\text{A, A, A}\}$

• Number of ways to arrange:

  $$\frac{3!}{3!}=1\text{ way}$$

Case 2: 2 letters are identical and 1 letter is different

We have two choices for the repeated pair: it can be a pair of As or a pair of Ns.

• Sub-case (i): Pair of A's $\{\text{A, A, \_}\}$

  • The remaining letter can be chosen from $\{\text{B, N}\}$ $\to {}^2{C}_1=2$ ways.

  • Arrangements for each selection (e.g., A, A, B): $\frac{3!}{2!}=3$ ways.

  • Total ways for this sub-case: $2\times 3=6$ ways.

• Sub-case (ii): Pair of N's $\{\text{N, N, \_}\}$

  • The remaining letter can be chosen from $\{\text{B, A}\}$ $\to {}^2{C}_1=2$ ways.

  • Arrangements for each selection (e.g., N, N, B): $\frac{3!}{2!}=3$ ways.

  • Total ways for this sub-case: $2\times 3=6$ ways.

$$\text{Total for Case 2}=6+6=12\text{ ways}$$

Case 3: All 3 letters are distinct

The distinct letters available are B, A, and N.

• Selection: $\{\text{B, A, N}\}$ $\to 1$ way.

• Arrangements:

  $$3!=3\times 2\times 1=6\text{ ways}$$

2. Calculate Total Number of Words:

Add the possibilities from all mutually exclusive cases together:

$$\text{Total Words}=\text{Case 1}+\text{Case 2}+\text{Case 3}$$

$$\text{Total Words}=1+12+6=19\text{ ways}$$

Correct Answer

The correct option is (c) 19.