How many different usernames are there if: a username must be a single capital l

Question

How many different usernames are there if: a username must be a single capital letter or a single digit (like "Z" or "0")? a username must be a single capital letter followed by a single digit (like "F0")? a username must be three capital letters followed by three digits (like "ABA101")? a username must consist of three capital letters and one digit (like "CB3C")? a username must consist of exactly five capital letters and exactly four digits (like "13ABC3B7Q")? Use the fact that C(n, k) = n!/(n-k)!k! to show that C(n, k) = C(n, n - k). There are 6 anagrams (rearrangements) of the word BAT. They are: ABT, ATB, BAT, BTA, TAB, TBA. (We include the word BAT itself. The anagrams need not be actual words.) How many ways can you anagram the following words? COMPUTER ANAGRAM [In class, we used a certain argument to show that C(7, 3) 3! = P(7, 3). Make a similar argument here. MATHEMATICS How many ways are there to seat five people around a circular table, where seatings are considered to be the same if they can be obtained from each other by rotating the table? Each letter (A-Z) is represented in Morse code by a sequence of signals (dots or dashes). Show that each letter can be assigned its own sequence of

Explanation / Answer

Number of capital letters = 26

Number of digits = 10

a)

Total number of different usernames = 26+10 = 36

b)

Total usernames = 26 x 10 = 260

c) L for letters and D for digit.

Total usernames = 26 x 26 x 26 x 10 x 10 x 10 = 17576000

d)

Let First digit and other three are letters: DLLL

Usernames when its DLLL= 10 *26*26*26 = 175760

Number of combination for one digit and 3 letters = 4C1 * 3C3 = 4*1 = 4

Total usernames= 4 * 175760 = 703040.

First Digit Second Digit Number of ways 26 10

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