If $^{43}{C_{r – 6}} = {\,^{43}}{C_{3r + 1}},$ then the value of $r$ is

The English alphabet has $5$ vowels and $21$ consonants. How many words with two different vowels and $2$ different consonants can be formed from the alphabet?

In how many ways can a student choose a programme of $5$ courses if $9$ courses are available and $2$ specific courses are compulsory for every student?

The number of ways in which thirty five apples can be distributed among $3$ boys so that each can have any number of apples, is

The number of words (with or without meaning) that can be formed from all the letters of the word $"LETTER"$ in which vowels never come together is