Question Type 6: Applying formula to disjoint cases Exercises

Find the number of bit strings of length 8 that contain at least 3 ones and at most 5 zeros.

Find the number of ways 5 balls can be selected such that at least 3 red balls are included.

How many 5-digit numbers with distinct digits can be formed using the digits $1$ to $7$ if the number contains at least two even digits?

In how many ways can 5 books be chosen from 7 novels and 5 textbooks if at least 2 are novels and at most 3 are textbooks?

From a group of 6 men and 8 women, how many ways can a committee of 5 be formed with at most 2 men?

In how many ways can you form a 4-person committee from 6 teachers and 7 parents with at most 2 parents?

A soccer team selects 11 players from 18 candidates: 8 defenders, 5 midfielders, and 5 forwards. How many selections have at least 3 defenders and at most 2 forwards?

From 5 boys and 4 girls, find the number of ways to form a committee of 4 with more girls than boys.

From 9 points, 4 points are chosen to form a quadrilateral (assuming no three points are collinear). How many quadrilaterals contain at least one of two special points $A$ and $B$?

How many ways can you choose 7 students from 10 girls and 8 boys so that the number of girls is between 3 and 5 inclusive?