If all the six digit numbers $x_1 x_2 x_3 x_4 x_5 x_6$ with $0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6$ are arranged in the increasing order, then the sum of the digits in the $72^{\text {th }}$ number is $............$.

All the five digits numbers in which each successive digit exceeds its predecessor are arranged in the increasing order of their magnitude. The $97^{th}$ number in the list does not contain the digit:

$^{47}{C_4} + \mathop \sum \limits_{r = 1}^5 {}^{52 - r}{C_3} = $

Let $n(A) = 3, \,n(B) = 3$ (where $n(S)$ denotes number of elements in set $S$), then number of subsets of $(A \times B)$ having odd number of elements, is-

A test consists of $6$ multiple choice questions, each having $4$ alternative ans wers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is

In a conference of $8$ persons, if each person shake hand with the other one only, then the total number of shake hands shall be