4) Looking at mathematics behind answers
The word Lucy has 4 letters l,u,c and y. If we know how many letters (no repeats) are in a word we can work out the number or permutations by multiplying 1 x 2 x 3 x 4 and so on until we reach the correct number of digits in a word (e.g. a 10 letter word would be 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10). The mathematical symbol for this is n! and stands for number factorial, this shows number of letters! . After working this out we can quickly and easily work out the number of permutations made by a word with repeats – this id done by dividing our previous answer by 2. For 2 repeats we divide the answer again by 2. This can be done as many times for the number or repeats.
Using patterns found to predict results
Conclusion
This investigation shows that the number of permutations (P) can be worked out by multiplying the number of letters (R) (including repeats) together. Each letter has a number designated to it – the first letter of the word will be 1 the second will be 2 the third will be 3 etc. The value of the letter depend on its position /in the word. No letter should get a higher value than the number of letters in the complete word.
The first number of permutations for your first word holds a clue to how many permutations you can get, as the size of the word grows bigger. In my investigation the first number of permutation for my 3-letter word was 6
If you divide the number permutations into the a word with one more letter number of permutations you get a pattern
(P)
6 ÷6= 1
24 ÷6= 4
120 ÷6= 20 ÷4 = 5
720 ÷6= 120 ÷20 = 6
5040 ÷6= 840 ÷120 = 7
40320 ÷6 = 6720 ÷ 840 = 8
362880 ÷6 = 60480 ÷6720 =9
Therefore the next number is 9 as it has 9 letters in the word 1x2x3x4x5x6x7x8x9 = permutations 362880. I predict if I divide it by 6 first and then by 6720 my answer should be 9. IT WAS.
Therefore if I try it with a word with 10 letters in my word 1x2x3x4x5x6x7x8x9x10
3628800 ÷ 6 =604800 and then I divide it by the number of permutations of my last word my answer should be 10. IT WAS
Rule: The total number of arrangements of n items with p items the same, q items the same and so on is:
n! / p! q! ...
(In words: n factorial over p factorial times q factorial and so on)
Example1
How many permutations can be formed from the letters, taken 5 at a time, of the word DADDY?
Solution (has 3 letters that are the same – D)
5! / 3! = 20
- The 5! Shows the word has five letters in it including repeats.
- The 3! Shows the word has 3 letters that are the same (d)
Example2
Find the number of arrangements of all the letters in the word MARMALADE.
Solution
9! / 2! x 3! = 30240
- The 9! Shows the word has 9 letters altogether including repeats.
- The 2! Shows the word had 2 letters the same (m).
- The 3! Shows the word has 3 more letters that are the same (a).
This formula can be used conclusively for other problems.
In how many ways can 3 apples, 2 oranges, 4 pears and one banana be given to 10 children if each child receives a piece of fruit?
Solution
10! / 3! x 2! x 4! = 12600
