JEE Main 2023 (Online) 1st February Evening ShiftQuestion 1
The total number of six digit numbers, formed using the digits $4,5,9$ only and divisible by $6$, is
Permutation and Combination
Answer$81$
Solution
Concept. Divisible by 6 = even and digit sum divisible by 3. Formulas used. (1) Remainders mod 3: $9\to0$, $4\to1$, $5\to2$Step 1 - Last digit. The only even digit is 4. Step 2 - First four digits. Any: $3^4=81$ ways. Step 3 - Fifth digit. Digits 9, 4, 5 have remainders $0,1,2$, one of each. So exactly one choice of the fifth digit makes the total sum divisible by 3. Step 4 - Total. $81\cdot1\cdot1=81$ (one third of $3^5$). Answer: 81