In how many ways can the letters of the word 'DIGEST' be arranged so that the vowels do not come together?

Study the following information to answer the given questions:

A committee of some members is to be made from a group of $8$men and $6$ women. How many different ways can it be made if the committee is to be made according to the following stipulation?

The committee of$4$ must include at least $1$woman.

A committee of some members is to be made from a group of $8$ Men and$6$ Women. How many ways can it be made if the committee is to be made according to the following stipulation?

The committee of $6$ Must-have exactly $3$ Men and $3$ Women?

Answer the questions on the basis of the following data:

A committee of $5$ members is to be formed by selecting out of $4$ men and $5$ women.How many different ways can the committee be formed if it should have at least $1$ man?

Answer the questions on the basis of the following data:

A committee of $5$ members is to be formed by selecting out of $4$ men and $5$ women.In how many different ways can the committee be formed if it should have $2$ men and at least $3$ women?