Problem 1 [26 pts (2,4,2,4,5,5,4)]: Unusual Security Each of the following descr

Question

Problem 1 [26 pts (2,4,2,4,5,5,4)]: Unusual Security Each of the following describes a possible scheme for setting passcodes for a door. Give the size of the password space - in other words, the number of possibilities. Show both where your numbers are coming from (eg. P(n, r) or C(n, r) if applicable) and a final numerical answer i. There are 8 buttons, and 4 of them must be pressed simultaneously. ii. There are 8 buttons on the door, and some nonempty subset of them must be pressed simul- taneously to open the door. iii. There are 20 buttons, and 4 of them must be pressed in a particular order, without repetition. iv. There are 20 buttons, and 4-6 of them must be pressed in a particular order, without repeti- tion v. There are 20 buttons, and 4 of them must be pressed simultaneously. But they can't all be in the same row or same column of the 4 x 5 keypad. vi. There are 20 buttons, and a passcode consists of hitting 3 of them simultanously, then a different set of 3, then a set of 3 that is different from those two. (The set is different if at least one button is different; the sets of buttons could overlap.) (Hint: You seem to be using up exactly one possibility each time. vii. The lock consists of 9 holes, and 5 marbles must be dropped into the right holes to open the door. Each hole needs the correct number of marbles for the door to open. Up to 4 marbles can fit in a single hole

Explanation / Answer

i. There are 8 buttons and 4 of them must be pressed simultaneously. This is equivalent to selecting 4 buttons out of 8. Hence, the number of possibilities is C(8,4) = 70

ii. There are 8 buttons and we need to press some non-empty subset of them simultaneously to open the door. This is equivalent to selecting any non-zero number of buttons out of 8 and pressing them simultaneously. Hence, the number of possibilities is equal to

C(8,1) + C(8,2) + C(8,3) + C(8,4) + C(8,5) + C(8,6)  + C(8,7) + C(8,8)

= 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1

= 255

iii. There are 20 buttons and 4 of them must be pressed in a particular order to open the door. This is equivalent to selecting 4 buttons out of 20 and then pressing them in a specified order. Hence, the number of possibilities is equal to C(20,4) * 4! = 4845 * 24 = 116280

iv. There are 20 buttons and 4-6 of them must be pressed in a particular order. This is equivalent to selecting either 4 or 5 or 6 buttons out of 20 and then pressing them in a specified order. Hence, just like the above case, the number of possibilities is equal to

C(20,4) * 4! + C(20,5) * 5! + C(20,6) * 6!

= 4845 * 24 + 15504 * 120 + 38760 * 720

= 29883960

v. There are 20 buttons and 4 of them must be pressed simultaneously. and they can be in the same row or same column of the 4*5 keypad. First we select the 4 columns out of 5 columns. This can be done in C(5,4) ways. Now, we select one button from each column one-by-one while also seeing to it that only one button per row is selected. The first button from the first column can be selected in 4 ways. The second button can be selected from the second column in 3 ways, so that it is not in the same row as the first button. The third button can be selected from the third column in 2 ways so that it is not in the same row as the first two buttons. There is only 1 way to select the 4th button from the fourth column.

Hence, the number of possibilities is C(5,4) * (4*3*2*1) = 120

vi. The first set can be selected in C(20,3) ways. A second set which is different than the first one can be selected in C(20,3) - 1 ways. The third set can be selected in C(20,3) - 2 ways. Therefore, total number of possibilities

= C(20,3) * [ C(20,3) - 1 ] * [ C(20,3) - 2 ]

= 1140 * 1139 * 1138

= 1477647480

vii. If 5 holes are needed, then each hole will contain 1 marble each. Therfore, we need to select 5 holes from 9. Hence, number of ways = C(9,5) = 126

If 4 holes are needed, then one of the holes will contain 2 marbles and rest will contain 1 marble each. Therefore, we first select 4 holes from 9 and then 1 out of these 4 to contain 2 marbles. Hence, number of ways = C(9,4) * C(4,1) = 126 * 4 = 504

If 3 holes are needed, we have two possibilities. Either

If 2 holes are needed, then we have two possibilities:

Therefore, total number of ways = 126 + 504 + (252 + 252) + (72 + 72) = 1278

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