Find the probability that a randomly selected arrangement of all the letters in the word $\mathrm{PALLETTE}$ begins and ends with the same letter.

Eight-digit mobile phone numbers issued by the Lemon Network all begin with $79.$

How many different phone numbers can the network issue?

Eight-digit mobile phone numbers issued by the Lemon Network all begin with $79.$

Find the probability that a randomly selected number issued by this network:

ends with the digits $97$

Eight-digit mobile phone numbers issued by the Lemon Network all begin with$79.$

Find the probability that a randomly selected number issued by this network:

reads the same left to right as right to left.

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

Find the number of different arrangements if there are no restrictions.

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

Find the number of different arrangements which start and finish with the letter $M.$

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 $2 Es.$