How many $2$-digit odd numbers can be formed from the digits $1, 2, 3, 4, 5$ and $8$,if repetition of digits is allowed?

Number of different words that can be formed from all letters of the word $APPLICATION$ such that two vowels never come together is -

In how many different ways can the letters of the word '$AUCTION$' be arranged in such a way that the vowels always come together?

$A$ question paper has $2$ parts,part $A$ and part $B$,each containing $10$ questions. If a student has to choose $8$ questions from part $A$ and $5$ questions from part $B$,in how many ways can the student choose the questions?

The total number of words that can be formed using the letters $a, b, c, d, e, f$ taken $3$ at a time such that each word contains at least one vowel is:

Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all five balls. In how many ways can we place the balls so that no box remains empty?