The total number of three-digit numbers,divisible by $3$,which can be formed using the digits $1, 3, 5, 8$,if repetition of digits is allowed,is:

In how many different ways can three persons $A, B, C$ having $6, 7$ and $8$ one-rupee coins respectively,donate $₹ 10$ collectively?

Let $5$-digit numbers be constructed using the digits $0, 2, 3, 4, 7, 9$ with repetition allowed,and are arranged in ascending order with serial numbers. Then the serial number of the number $42923$ is $...............$.

$A$ five-digit number divisible by $3$ has to be formed using the numerals $0, 1, 2, 3, 4,$ and $5$ without repetition. The total number of ways in which this can be done is:

The number of $4$-letter permutations formed using the English alphabet such that the number of distinct vowels is equal to the number of distinct consonants,when repetition is allowed,is

Let ${}^nC_{r-1}=28$,${}^nC_r=56$,and ${}^nC_{r+1}=70$. Let $A(4 \cos t, 4 \sin t)$,$B(2 \sin t, -2 \cos t)$,and $C(3r - n, r^2 - n - 1)$ be the vertices of a triangle $ABC$,where $t$ is a parameter. If $(3x - 1)^2 + (3y)^2 = \alpha$ is the locus of the centroid of triangle $ABC$,then $\alpha$ equals: