The $11$ letters of the word REMEMBRANCE are arranged in a line.

Find the number of different arrangements which do not have all $4$ vowels $(E, E, A, E)$ next to each other. 

The $11$ letters of the word REMEMBRANCE are arranged in a line.

$4$ letters from the letters of the word REMEMBRANCE are chosen.

Find the number of different selections which contain no$Ms$ and no $Rs$ and at least $2 Es.$

Find how many ways $15$ children can be divided into three groups of five if:

there are no restrictions

Find how many ways $15$ children can be divided into three groups of five if:

two of the children are brothers who must be in the same group.

An entertainer has been asked to give a performance consisting of four items. They know three songs, five jokes, two juggling tricks and can play one tune on the mandolin. Find how many different ways there are for them to choose the four items if:

there are no restrictions on their performance

An entertainer has been asked to give a performance consisting of four items. They know three songs, five jokes, two juggling tricks and can play one tune on the mandolin. Find how many different ways there are for them to choose the four items if:

they decide not to sing any songs

An entertainer has been asked to give a performance consisting of four items. They know three songs, five jokes, two juggling tricks and can play one tune on the mandolin. Find how many different ways there are for them to choose the four items if:

they are not allowed to tell more than two jokes.