SOLUTION: The surnames of 40 students in a class were arranged in alphabetical order. 16 of the surnames begin with O while 9 of the surnames begin with A. 14 of the letters of the alphabe

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Question 1210699: The surnames of 40 students in a class were arranged in alphabetical order. 16 of the surnames begin with O
while 9 of the surnames begin with A. 14 of the letters of the alphabet do not appear as the first letter of the
surname.
(a). What is the probability that the surname of a child picked at random from the class begins with either A or O?
(b). If there is only one letter besides A and O, such that more than one surname begins with this letter,
how many surnames begin with that letter?

Answer by ikleyn(54017) About Me  (Show Source):
You can put this solution on YOUR website!
.
The surnames of 40 students in a class were arranged in alphabetical order. 16 of the surnames begin with O
while 9 of the surnames begin with A. 14 of the letters of the alphabet do not appear as the first letter of the
surname.
(a). What is the probability that the surname of a child picked at random from the class begins with either A or O?
(b). If there is only one letter besides A and O, such that more than one surname begins with this letter,
how many surnames begin with that letter?
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Question (a) is very simple. The probability under this question is  %2816%2B9%29%2F40 = 25%2F40 = 5%2F8.


Question (b) requires longer reasoning.


After excluding 16 students whose surnames start with A and excluding 9 students whose surnames start with O,
we have 40 - 16 - 9 -= 40 - 25 = 15 remaining students and 15 remaining surnames.


For them, we have 26 - 14 - 2 = 10 letters that really are the first letters of their surnames.

Of these 10 letters, only one is repeated, as given in the problem. The remaining 9 letters are unique.


Hence, the repeated letter appears 15 - 9 = 6 times as the first letter of surnames besides A and O.


At this point, the problem is solved completely.


ANSWER. How many surnames begin with that letter? 6 (six) surnames.