JEE Main 2019 (Online) 10th April Morning ShiftQuestion 1
The number of 6 digit numbers that can be formed using the digits $0,1,2,5,7$ and $9$ which are divisible by $11$ and no digit is repeated is :
Permutation and Combination
Solution
Concept. All six digits are used. Divisible by $11$ means (sum of odd places) $-$ (sum of even places) is a multiple of $11$. Formulas used. (1) $11\mid N\iff S_{\text{odd}}-S_{\text{even}}\equiv0\pmod{11}$(2) Multiplication principleStep 1 - Total. $0+1+2+5+7+9=24=S_o+S_e$. Step 2 - Difference. $S_o-S_e$ is even (same parity as $24$) and $|S_o-S_e|\le 21-3=18$. So $S_o-S_e=0$, $S_o=S_e=12$. Step 3 - Split. Three digits with sum $12$: $\{0,5,7\}$ and its complement $\{1,2,9\}$. This is the only split. Step 4 - $\{0,5,7\}$ in odd places (1st, 3rd, 5th). First digit not $0$: $2\cdot2\cdot1=4$; even places $3!=6$; $4\cdot6=24$. Step 5 - $\{1,2,9\}$ in odd places. $3!\cdot3!=36$. Step 6 - Add. $24+36=60$. Answer: B