Binomial expansionsP1^
3
SOlUTION
(i) 5
05
05 0
120
1 120 1
= !( −! )!= × =
(ii) 5
15
14
120
124
5
==
×
! =
!!
(iii) 5
25
23
120
26
10
==
×
! =
!!
(iv) 5
35
32
120
62
10
==
×
! =
!!
(v) 5
45
41
120
24 1
5
==
×
! =
!!
(vi) 5
55
50
120
120 1
1
==
×
! =
!!
Note
You can see that these numbers, 1, 5, 10, 10, 5, 1, are row 5 of Pascal’s triangle.Most scientific calculators have factorial buttons, e.g. x!. Many also have nCr
buttons. Find out how best to use your calculator to find binomial coefficients, as^
well as practising non-calculator methods.ExamPlE 3.15 Find the coefficient of x^17 in the expansion of (x + 2)^25.
SOlUTION(x + 2)^25 = 25
0
x^25 + 25
1
^ x(^24 21) + 25
2
^ x(^23 22) + ... + 25
8
x^17 28 + ...^25
25
225
So the required term is^25
8
× 28 × x^1725
8
25
817
25 24 23 22 21 20 19 18 17
8
==! ×× ×× × ×××
!!
!
!!!× 17
= 1 081 575.
So the coefficient of x^17 is 1 081 575 × 28 = 276 883 200.Note
Notice how 17! was cancelled in working out ^258 . Factorials become large numbers
very quickly and you should keep a look-out for such opportunities to simplify
calculations.