Permutations and Combinations – Word Formation – JEE Main 23 Jan 2026 Shift 1

Solution:

List the frequency of letters in “PQRPQRSTUVP”:
Total letters = 11.
P: 3
Q: 2
R: 2
S: 1
T: 1
U: 1
V: 1

We need to form 4-letter words. We split into cases based on the selection of letters.

**Case 1: 3 alike, 1 different**
– Alike letters can be selected from {P} (1 way).
– The different letter can be selected from the remaining 6 types {Q, R, S, T, U, V} (6 ways).
– Arrangement: $\frac{4!}{3!} = 4$ ways.
– Total = $1 \times 6 \times 4 = 24$.

**Case 2: 2 alike, 2 alike**
– Pairs can be selected from {P, Q, R} (select 2 types out of 3): $^3C_2 = 3$ ways.
– Arrangement: $\frac{4!}{2!2!} = 6$ ways.
– Total = $3 \times 6 = 18$.

**Case 3: 2 alike, 2 different**
– One pair from {P, Q, R} (3 ways).
– Two different letters from the remaining 6 types: $^6C_2 = 15$ ways.
– Arrangement: $\frac{4!}{2!} = 12$ ways.
– Total = $3 \times 15 \times 12 = 540$.

**Case 4: All 4 different**
– Select 4 letters from 7 distinct types {P, Q, R, S, T, U, V}: $^7C_4 = 35$ ways.
– Arrangement: $4! = 24$ ways.
– Total = $35 \times 24 = 840$.

**Total Number of Words:**
$$ Total = 24 + 18 + 540 + 840 $$
$$ Total = 1422 $$

Ans. 1422