The total number of 5 -digit numbers, formed by using the digits 1, 2, 3, 5, 6, 7 without repetition, which are multiple of 6 , is

$$\begin{gathered} \text{ To make a no. divisible by }3\text{ we can use the digits either }1,2,5,6,7 \mathrm{or} 1,2,3,5,7 \\ \text{ Using }1,2,5,6,7\text{, number of even numbers is }=4\times 3\times 2\times 1\times 2=48 \\ \text{ Using }1,2,3,5,7\text{, number of even numbers is }=4\times 3\times 2\times 1\times 1=24 \\ \mathrm{Total} \mathrm{number} \mathrm{of} \mathrm{required} \mathrm{numbers}=48+24=72 \end{gathered}$$