I'm having trouble with this question.
a appears in the first position 50388 times
b appears in the first position 31824 times and in the second 18564 times etc,
so in order to calculate a
i did $19C7$
for b: it was $18C7$ for the first position and $18C6$ for the second
for c I did $17C7$ and got 19448 but $17C6$ or $17C5$ didn't work
What am I doing wrong?
I know all the letters appear 50388 times but how can i explain this without adding each individual total
i think if i understand part a then i can do part b
Your set of words is all the ${20 \choose 8}=125\ 970$ words by taking $8$ letters out of the $20$ without replacement and keeping the letters in alphabetical order. $a$ always appears in first position if it is present because nothing else comes before. $b$ appears first if $a$ is not present, so you need to choose $7$ letters out of the remaining $18$ to complete the word. If $a$ and $b$ are both present, you have to choose $6$ out of the remaining $18$ to complete the word. For $c$, if it is in first position $a$ and $b$ are missing, so you need to choose $7$ of the remaining $17$ letters to complete the word. If it is in second position, exactly one of $a$ or $b$ is present.