Question 321941
a)There are 26 letters (A-Z), 10 digits (0-9), 5 vowels (A,E,I,O,U)
We'll have to break up the count to include when the first letter is not a vowel and when the first letter is a vowel. 
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When the first letter is not a vowel.
First slot: Letter - 21 choices
Second slot: Number - 10 choices
Third slot: Number - 9 choices
Fourth slot: Number - 8 choices
Fifth slot: Number -7 choices
Sixth slot: Number - 6 choices
Seventh slot: Vowel - 5 choices
Eigth slot: Vowel - 4 choices
{{{N1=(21*10!)/3!}}}
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When the first letter is a vowel.
First slot: Letter - 5 choices
Second slot: Number - 10 choices
Third slot: Number - 9 choices
Fourth slot: Number - 8 choices
Fifth slot: Number -7 choices
Sixth slot: Number - 6 choices
Seventh slot: Vowel - 4 choices
Eigth slot: Vowel - 3 choices
{{{N2=5*10*9*8*7*6*4*3=10!/2!}}}
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{{{Ntot=N1+N2}}}
{{{Ntot=(21*10!)/3!+10!/2!}}}
{{{Ntot=10!(21/6+3/6)}}}
{{{Ntot=4*10!=14515200}}}
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b) 
First slot: Letter - 26 choices
Second slot: Number - 10 choices
Third slot: Number - 9 choices
Fourth slot: Number - 8 choices
Fifth slot: Number -7 choices
Sixth slot: Number - 6 choices
Seventh slot: Vowel - 5 choices
Eigth slot: Vowel - 5 choices
{{{N=26*10*9*8*7*6*5*5}}}
{{{N=19656000}}}