The number of students in classes of higher secondary schools of a city follows normal distribution. Average number of students in the classes is $50$ and standard deviation is $15 .$ If a class is selected at random, then find the following probabilities $(1)$ a class consists of more than $68$ students. $(2)$ a class consists of less than $32$ students.
$(A)$ For a normal distribution, the first quartile and the mean deviation are $20$ and $24$ respectively. Obtain an estimate of the value of mode.
$(B)$ An amount of purchase of a customer in a Mall of a city follows normal distribution with mean $₹ 800$ and standard deviation $₹ 200$. If a customer is selected at random, then find the probability that the amount of purchases made by him is in between $₹ 850$ to $₹ 1,200$.