A certain website requires user to set up a login password consisting of four di

Question

A certain website requires user to set up a login password consisting of four digits. Each digit can be selected from 0 9. Repetitions are allowed. (a) How many different passwords are possible if there are so other restrictions on the choice of digits? (b) Suppose the following choices of digits for the password are. Determine the number of passwords which is NOT allowed in each of the following (i) All four digits are identical. (ii) Sequence of ascending or descending digits, such as (iii) Any sequence starting with 19 or 20 birth years which are

Explanation / Answer

We are considering a login password which is n = 4 numericals long, with each digit between 0-9.

(a) If there is no restriction on the choice of digits, each digit can be selected in 10 different ways and thus four of the digits can be selected in 104 different ways.

(b) (i) If all the four digits can't be identical, we can't have 10 of the following choices: 0000, 1111, 2222, 3333, 4444, 5555, 6666,7777,8888,9999.

(ii) Consecutive ascending digits can be, starting with 0:

"0123" "0124" "0125" "0126" "0127" "0128" "0129" "0134" "0135" "0136" "0137" "0138" "0139" "0145" "0146" "0147" "0148" "0149" "0156" "0157" "0158" "0159" "0167" "0168" "0169" "0178" "0179" "0189" "0234" "0235" "0236" "0237" "0238" "0239" "0245" "0246" "0247" "0248" "0249" "0256" "0257" "0258" "0259" "0267" "0268" "0269" "0278" "0279" "0289" "0345" "0346" "0347" "0348" "0349" "0356" "0357" "0358" "0359" "0367" "0368" "0369" "0378" "0379" "0389" "0456" "0457" "0458" "0459" "0467" "0468" "0469" "0478" "0479" "0489" "0567" "0568" "0569" "0578" "0579" "0589" "0678" "0679" "0689" "0789"

which is 84 cases,

with 1:

"1234" "1235" "1236" "1237" "1238" "1239" "1245" "1246" "1247" "1248" "1249" "1256" "1257" "1258" "1259" "1267" "1268" "1269" "1278" "1279" "1289" "1345" "1346" "1347" "1348" "1349" "1356" "1357" "1358" "1359" "1367" "1368" "1369" "1378" "1379" "1389" "1456" "1457" "1458" "1459" "1467" "1468" "1469" "1478" "1479" "1489" "1567" "1568" "1569" "1578" "1579" "1589" "1678" "1679" "1689" "1789"
we have 56 cases,

with 2:

"2345" "2346" "2347" "2348" "2349" "2356" "2357" "2358" "2359" "2367" "2368" "2369" "2378" "2379" "2389" "2456" "2457" "2458" "2459" "2467" "2468" "2469" "2478" "2479" "2489" "2567" "2568" "2569" "2578" "2579" "2589" "2678" "2679" "2689" "2789"

35 cases,

with 3

"3456" "3457" "3458" "3459" "3467" "3468" "3469" "3478" "3479" "3489" "3567" "3568" "3569" "3578" "3579" "3589" "3678" "3679" "3689" "3789"
20 cases,

with 4,

"4567" "4568" "4569" "4578" "4579" "4589" "4678" "4679" "4689" "4789"

10 cases,

with 5,

"5678" "5679" "5689" "5789"
4 case,

and 6

"6789"

1 case, and thus overall we have 210 cases.

With descending, we have, by symmetry, 210 more cases. Thus, overall, we have 420 case.

(iii) With 19 and 20, we have 100 possible cases each, and thus overall 200 cases.

(c) Each of the three cases, have distinct events favoring them. Thus, the number of cases not favoring the password is 10+420+200 = 630 cases.

The probability of a randomly chosen password being a valid password is (10000-630)/10000 = 0.937.

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