IB HL Counting Binomial Questions

IB Math HL Counting Principle and Binomial Expansions

1.

Mr Blue, Mr Black, Mr Green, Mrs White, Mrs Yellow and Mrs Red sit around a circular table for a meeting. Mr Black and Mrs White must not sit together. Calculate the number of different ways these six people can sit at the table without Mr Black and Mrs White sitting together.

2.

In how many ways can six different coins be divided between two students so that each student receives at least one coin?

3.

How many four-digit numbers are there which contain at least one digit 3?

4.

A committee of four children is chosen from eight children. The two oldest children cannot both be chosen. Find the number of ways the committee may be chosen.

5.

A team of five students is to be chosen at random to take part in a debate. The team is to be chosen from a group of eight medical students and three law students. Find the probability that

6.

(a)

only medical students are chosen;

(b)

all three law students are chosen.

There are 10 seats in a row in a waiting room. There are six people in the room. (a)

In how many different ways can they be seated?

(b)

In the group of six people, there are three sisters who must sit next to each other. In how many different ways can the group be seated?

7.

Twelve people travel in three cars, with four people in each car. Each car is driven by its owner. Find the number of ways in which the remaining nine people may be allocated to the cars. (The arrangement of people within a particular car is not relevant).

8.

Given that (1 + x)5 (1 + ax)6 ≡ 1 + bx + 10x2 + ............... + a6 x11, find the values of a, b ∈

*.

9.

Find the coefficient of x7 in the expansion of (2 + 3x)10, giving your answer as a whole number.

10.

 1  7 x + 2  ax   is 3 . Find the possible values of a. The coefficient of x in the expansion of

11.

 1  1 – x  Find the coefficient of x3 in the binomial expansion of  2  .

12.

(a)

Write down the term in xr in the expansion of (x + h)n, where 0 ≤ r ≤ n, n∈

(b)

Hence differentiate xn, n∈

7

8

+

.

+

, from first principles. 1

(c)

13.

Starting from the result xn × x–n = 1, deduce the derivative of x–n, n∈

+

.

Determine the first three terms in the expansion of (1− 2x)5 (1+ x)7 in ascending powers of x.

2