The sum of all $4$-digit numbers that can be formed by using the digits $2, 4, 6, 8$ (repetition of digits not allowed) is

The letters of the word $ARRANGE$ are used to form different words. All these words are arranged in dictionary order. What is the rank of the word $ARRANGE$ in the dictionary?

Let $b_{1} b_{2} b_{3} b_{4}$ be a $4$-element permutation with $b_{i} \in \{1, 2, 3, \ldots, 100\}$ for $1 \leq i \leq 4$ and $b_{i} \neq b_{j}$ for $i \neq j$,such that either $b_{1}, b_{2}, b_{3}$ are consecutive integers or $b_{2}, b_{3}, b_{4}$ are consecutive integers. Find the number of such permutations.

How many ways are there to arrange the letters of the word $EDUCATION$ so that all the following three conditions hold?
- The vowels occur in the same order $(E, U, A, I, O)$.
- The consonants occur in the same order $(D, C, T, N)$.
- No two consonants are next to each other.

Ten persons,amongst whom are $A, B$ and $C$,are to speak at a function. The number of ways in which it can be done if $A$ wants to speak before $B$ and $B$ wants to speak before $C$ is:

The number of five-digit numbers,greater than $40000$ and divisible by $5$,which can be formed using the digits $0, 1, 3, 5, 7,$ and $9$ without repetition,is equal to: