Week 7 - Counting 1

Friday, 5 April 2019

9:58 PM

How many ways can you choose r distinct objects from n possible objects? The answer depends upon whether the order of selection is important. • Ifthe order of selection is important, then the number ofways to choose r objects from n is called a permutation P(n, r) • Ifthe order of selection is not important, then the number ofways to choose r objects from n is called a combination C(n, r) The NUMBASsyntax for P(n, r) and C(n, r) is and . a) You have 8 disctinct Discrete Mathematics textbooks, but you can only display 7 on your bookshelf. How many ways can you arrange your Discrete Mathematics textbooks? Show steps (Your score will not be affected.) Answer: perm (8, 7) P7 Submit part Your answer is numerically correct. You were awarded 2 marks. You scored 2 marks forthis part. Score: 2/2 Answered b) You want to keep your valuable Discrete Mathematics textbooks close at all times. But it's time for class and you only have room for 7 Discrete Mathematics books in your backpack. How many ways can you pack your backpack? Show steps (Your score will not be affected.) Answer. comb (8, 7) C7

 

A string of 7 digits is made radmonly using only the non-zero digits 1, 2, 3, 4, 5, 6, 7, 8, 9. When each digit is chosen independently, the number of ways to choose the string is just the product of the number of ways to choose each of the digits. ways to choose i'th digit). # of ways to choose string a) Assuming there are no other restrictions, how many strings are possible? Show steps (Your score will not be affected.) Submit part You revealed the steps. Your answer is numerically correct. You were awarded 2 marks. You scored 2 marks forthis part. Score: 2/2 Answered b) How many strings are possible if the first digit must be even, the second digit odd, and the remaining 5 digits are less than 7. Show steps (Your score will not be affected.) 4x5 x 65 Answer.

 

c) How many strings are possible which contain exactly 5 nines? Show steps (Your score will not be affected.) Answer: x 82 Submit part You revealed the steps. Your answer is numerically correct. You were awarded 2 marks. Because you received full marks for the part, your answers to the steps aren't counted. You scored 2 marks for this pan. Score: 2/2 Answered

 

Suppose you have a hypothetical new alphabet with 21 letters: 15 consonants and 6 vowels. Note that the NUMBAS syntax for C(n, r) is . a) How many different 5-letterwords can you make with this alphabet? 215 b) Fill out the following table 1 2 3 4 5 TOTAL Submit Your answer is numerically correct. You were awarded 0.5 You scored 0.5 marks forthis part. Score: 0.5/02 Answered Number of 5-letterwords with n vowels 155 comb ( S , comb (S , comb (S , comb (S , comb (S , 1) x 6 x 154 5C2 x 62 x 153 ca x 63 x 152 x 151 21 AS 215

 

c) How many 5-letter words have at least one vowel? Show steps (Your score will not be affected.) 5 Answer: 21 AS-ISAS 21 —155 Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered

 

You are playing a branded word game in which you are given 8 tiles with the letters How many 4 letter strings (not necessarily words from the English dictionary) can be made from these letters, if no tile is to be used more than once? The tricky part here is what to do about the repeated letter m. Let's consdier three cases where the letter occurs zero, one or two times. Then add up all these ways together to form the total number of words. How many 4 letter strings can be made, without using the letter m? perm (6, 4) Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered How many 4 letter strings can be made, that use the letter m exactly once? It might be helpful to first choose a position for m and then choose the remaining letters. comb (4, 1) *perm (6, 3) Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part.

 

How many 4 letter strings can be made, that use the letter m exactly twice? It might be helpful to first choose the positions for m and then choose the remaining letters. comb (4, 2) *perm (6, 2) Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered There is a more efficient way to answer this question, which involves multinomials. We shall visit this concept later. (Your score will not be affected.) Hide steps Answer: perm (6, 4) *comb (4, 1) *perm (6, 3) +comb (4, 2) *perm (6, 2) 6P4 +4 +4

 

The inclusion/exclusion principle can be stated as a relationship between the size of the union and intersection of sets. Suppose you have sets A and B, then IA u Bl = IAI + IBI — 131. This is a useful concept for counting. a) In a panicularyear, • 391 students study Discrete Mathematics, • 472 students study Computing IA, and • 227 students study both Discrete Mathematics and Computing IA. How many students study Discrete Mathematics or Computing IA? 636 Submit part Your answer is numerically correct. You were awarded 1 mark. You scored 1 mark for this part. Score: 1/1 Answered

 

b) In a panicularyear, 391 students study Discrete Mathematics, 472 students study Computing IA, 497 students study Physics IA, 227 students study both Discrete Mathematics and Computing IA, 162 students study both Discrete Mathematics and Physics IA, 166 students study both Computing IA and Physics IA, and 160 students study Discrete Mathematics, Computing IA and Physics IA. How many students study Discrete Mathematics or Computing IA or Physics IA? Show steps (Your score will not be affected.) 965 Submit part Your answer is numerically correct. You were awarded 5 marks. You scored 5 marks forthis part. Score: 5/5 Answered

 

Created with Microsoft OneNote 2016.