Available online
Description
The objective of the programme is to teach the Binomial Theorem in terms of numbers of choices. The programme starts with a particular case of the Binomial Theorem expanding (a+b)6. An animation sh...ows the mechanism of multiplying together six copies of the bracket (a+b). But rather than working through this in detail, the programme takes another view of raising (a+b) to a power. Models of a square and a cube, each with side (a+b), show how geometrical considerations lead to the expansions for (a+b)2 and (a+b)3 . Although geometrical intuition cannot be extended to a six-dimensional cube, the principles of choosing either a or b from each bracket are valid for higher powers. The problem then becomes: how many ways can you choose (for example) 2 out of 6? The answer to this problem is found in the city of Edinburgh. The street layout of Edinburgh contains a rectangular grid pattern with some streets running north/south and other streets running east/west. How may different ways are there of walking from one part of the grid to another? In the example chosen, this is the same as asking how many ways two south decisions can be made out of six possible choices. So the problem is the same as in the Binomial Theorem, but the street problem has the advantage of being pictorial and so the number of choices can be computed comparatively easily. The rectangular grid can be extended to a triangular form to compute all the coefficients in the expansion of (a+b)6, and the result is known as Pascal's Triangle. In principle, Pascal's Triangle can be constructed for any power of (a+b), but for high powers a symbolic method becomes essential. The symbol used is nCr, which means the number of ways of choosing r things out of n, and the calculations can be written.
The objective of the programme is to teach the Binomial Theorem in terms of numbers of choices. The programme starts with a particular case of the Binomial Theorem expanding (a+b)6. An animation sh...ows the mechanism of multiplying together six copies of the bracket (a+b). But rather than working through this in detail, the programme takes another view of raising (a+b) to a power. Models of a square and a cube, each with side (a+b), show how geometrical considerations lead to the expansions for (a+b)2 and (a+b)3 . Although geometrical intuition cannot be extended to a six-dimensional cube, the principles of choosing either a or b from each bracket are valid for higher powers. The problem then becomes: how many ways can you choose (for example) 2 out of 6? The answer to this problem is found in the city of Edinburgh. The street layout of Edinburgh contains a rectangular grid pattern with some streets running north/south and other streets running east/west. How may different ways are there of walking from one part of the grid to another? In the example chosen, this is the same as asking how many ways two south decisions can be made out of six possible choices. So the problem is the same as in the Binomial Theorem, but the street problem has the advantage of being pictorial and so the number of choices can be computed comparatively easily. The rectangular grid can be extended to a triangular form to compute all the coefficients in the expansion of (a+b)6, and the result is known as Pascal's Triangle. In principle, Pascal's Triangle can be constructed for any power of (a+b), but for high powers a symbolic method becomes essential. The symbol used is nCr, which means the number of ways of choosing r things out of n, and the calculations can be written.
| Module code and title: | M101, Mathematics: a foundation course |
|---|---|
| Item code: | M101; 02; 1982 |
| First transmission date: | 1982 |
| Published: | 1982 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:24:00 |
| + Show more... | |
| Producer: | David Saunders |
| Contributors: | Norman Gowar; John Mason |
| Publisher: | BBC Open University |
| Keyword(s): | Binominal Theory; Cube; Edinburgh street lay-out; Pascal's Triangle |
| Master spool number: | HOU4011 |
| Production number: | FOUM137H |
| Videofinder number: | 2427 |
| Available to public: | no |
