document.write( "Question 1201457: A security code is to be formed by using three alphabets and four digits chosen
\n" ); document.write( "from the alphabets {a,b,c,d,e} and digits {1,2,3,4,5,6}. All the digit and
\n" ); document.write( "alphabets can only be used one.Find the probability the number of different ways
\n" ); document.write( "of the security code can be formed if \r
\n" ); document.write( "\n" ); document.write( "i)all alphabets are next to each other and all digits are next to each other\r
\n" ); document.write( "\n" ); document.write( "ii)it consists at least two consonants \n" ); document.write( "
Algebra.Com's Answer #835903 by Edwin McCravy(20056)\"\" \"About 
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A security code is to be formed by using three alphabets and four digits chosen
\n" ); document.write( "from the alphabets {a,b,c,d,e} and digits {1,2,3,4,5,6}. All the digit and
\n" ); document.write( "alphabets can only be used one.Find \"cross%28the%29\"\"cross%28probability%29\" the number of different ways
\n" ); document.write( "of the security code can be formed if \r
\n" ); document.write( "\n" ); document.write( "i)all alphabets are next to each other and all digits are next to each other
[Examples: 5126eab, dce4325]\r\n" );
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document.write( "There are P(5,3) = (5)(4)(3) = 60 ways to arrange 3 alphabet letters.\r\n" );
document.write( "There are P(6,4) = (6)(5)(4)(3) = 360 ways to arrange 4 digits.\r\n" );
document.write( "There are P(2,2) = 2! = 2 ways to choose whether to put the alphabet letters or\r\n" );
document.write( "the digits first.\r\n" );
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document.write( "That's (60)(360)(2) = 43200 possible security codes for i).
\r
\n" ); document.write( "\n" ); document.write( "ii)it consists at least two consonants.
[Examples: 4d61ba2, c13d54b]\r\n" );
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document.write( "{a,b,c,d,e} contains only 3 consonants {b,c,d} and 2 vowels {a,e}. \r\n" );
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document.write( "There are two cases to choose the letters to use, exactly 2 consonants and 1\r\n" );
document.write( "vowel or all 3 consonants.  Then we'll choose the digits, then arrange the 7\r\n" );
document.write( "chosen characters in 7! ways.\r\n" );
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document.write( "Case 1. Choose exactly 2 consonants and 1 vowel. \r\n" );
document.write( "Choose the 2 consonants C(3,2) = 3 ways.\r\n" );
document.write( "Choose the 1 vowel in C(2,1) = 2 ways\r\n" );
document.write( "That's (3)(2) = 6 ways\r\n" );
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document.write( "Case 2. Choose all 3 consonants.\r\n" );
document.write( "That's just C(3,3) = 1 way.\r\n" );
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document.write( "That's a total of 6+1 = 7 ways to choose the alphabet letters to use.\r\n" );
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document.write( "There are C(6,4) = 15 ways to choose the 4 digits.\r\n" );
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document.write( "Now we can arrange each of the (7)(15) = 105 sets of 7 characters in\r\n" );
document.write( "P(7,7) = 7! = 5040 ways.\r\n" );
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document.write( "Answer: (105)(5040) = 529200 security codes with at least 2 consonants.\r\n" );
document.write( "Edwin
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