Binomial Theorem and Counting Principles [61 marks]
1.
[Maximum mark: 7]
23M.2.SL.TZ1.6
8
The coefficient of x6 in the expansion of (ax3 + b) is 448.
The coefficient of x6 in the expansion of (ax3 + b)
10
is 2880.
Find the value of a and the value of b, where a , b > 0.
2.
[7]
[Maximum mark: 16]
22N.1.AHL.TZ0.11
Consider a three-digit code abc, where each of a, b and c is assigned one of the
values 1, 2, 3, 4 or 5.
Find the total number of possible codes
(a.i)
(a.ii)
assuming that each value can be repeated (for example, 121 or
444).
[2]
assuming that no value is repeated.
[2]
Let P (x) = x3 + ax2 + bx + c, where each of a, b and c is assigned one
of the values 1, 2, 3, 4 or 5. Assume that no value is repeated.
Consider the case where P (x) has a factor of (x2 + 3x + 2).
(b.i)
Find an expression for b in terms of a.
[6]
(b.ii)
Hence show that the only way to assign the values is
a = 4, b = 5 and c = 2.
[2]
(b.iii) Express P (x) as a product of linear factors.
(b.iv) Hence or otherwise, sketch the graph of y = P (x), clearly
showing the coordinates of any intercepts with the axes.
[1]
[3]
3.
[Maximum mark: 6]
(a)
Write down and simplify the first three terms, in ascending
powers of x, in the Extended Binomial expansion of (1 − x)
.
(b)
4.
By substituting x = 9 find a rational approximation to √9.
1
3
[Maximum mark: 4]
Find the coefficient of x8 in the expansion of (2x − 5) .
[Maximum mark: 4]
9
[Maximum mark: 4]
[3]
[3]
[4]
[4]
24N.2.AHL.TZ0.2
9
Find the coefficient of x6 in the expansion of (2x − 5) .
7.
3
24N.2.SL.TZ2.2
Find the coefficient of x6 in the expansion of (2x − 5) .
6.
1
24N.2.SL.TZ1.2
11
5.
EXM.2.AHL.TZ0.1
[4]
[Maximum mark: 6]
24N.2.AHL.TZ0.7
A self-service sushi restaurant has a row of 12 available seats, as shown in the
following diagram.
Anvi, Vanya and Parita decide to go to the restaurant for lunch.
(a)
Find the number of possible ways that they can be seated in
this row, if they decide not to sit together as a group of 3.
[3]
The next day, Anvi, Vanya and Parita are joined by 3 additional people in the
same restaurant, and they sit in the same row of 12 available seats. Anvi, Vanya
and Parita now decide to sit next to each other as a group of 3.
(b)
8.
Find the number of possible ways that these 6 people can be
seated.
[3]
[Maximum mark: 7]
24M.1.AHL.TZ2.9
A teacher takes n students on a field trip. The students are assigned randomly
into two groups.
For safety reasons there must be exactly three students in the first group and at
least three students in the second group.
The teacher will randomly assign three students to the first group and the other
students to the second group.
(a)
Write down an expression for the number of ways that the
students could be assigned.
Two of the students ask the teacher not to work in the same group.
The teacher agrees and now finds that the number of ways to assign the students
is halved.
[1]
(b)
9.
Determine the value of n.
[6]
[Maximum mark: 7]
24M.2.AHL.TZ1.9
A group of 10 children includes one pair of brothers, Alvin and Bobby, and one
pair of sisters, Catalina and Daniela.
The children are to be seated at 10 desks which are arranged in two rows of five
as shown in the following diagram.
Alvin and Bobby must be seated next to each other in the same row.
(a)
Find the total number of ways the children can be seated.
[3]
After an argument, Catalina and Daniela must not be seated next to each other.
Alvin and Bobby must still be seated next to each other.
(b)
Find the total number of ways the children can be seated.
© International Baccalaureate Organization, 2025
[4]