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Counting the Number of Seating Arrangements
Problem 103: A family has 12 members. In how many ways can six family members be seated in a row so that their ages increase from left to right? Solution: Since the order in which we seat the family members matter, this is a combination problem. That means that we can seat the family members…
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Counting the Number of License Plates
Problem 102: Suppose that license plates from a certain state consist of four different letters followed by three different digits. In how many license plates are the letters in alphabetical order and the digits in increasing order? Solution: We can think of this problem as having 7 slots (4 for letters, 3 for digits) available…
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Counting the Number of Photo Session
Problem 101: There are three women and six men on the United States Supreme Court. In how many ways can the justices be seated in a row for a group picture in which the three women sit next to each other. Solution: one can do this problem steps by steps. First, we can seat the…
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Counting the Number of Displaying Paintings
Problem 100: An art gallery has five paintings, by each of three artists, hanging in a row, with paintings by the same artist grouped together. How many different arrangements are possible? Solution: We will solve this problem steps by steps. First, we will arrange the artist. We have three artists. We can arrange them ways….
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Counting the Number of Palindrome (Alphabet)
Problem 99: Palindromes: A number or word is said to be a palindrome if it reads the same backwards and forwards (e.g., 58485 or radar). How many 4-letter words (including nonsense words) are palindrome? Solution: One can think of this problem as having four available slots to fill up. The five spots available are the…
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Counting the Number of Palindrome (Digits)
Problem 98: Palindromes: A number or word is said to be a palindrome if it reads the same backwards and forwards (e.g., 58485 or radar). How many 5-digit numbers are palindromes? Solution: One can think of this problem as having five available slots to fill up. The five spots available are the spot for each…
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One-Sided-Limit II
Problem 97: Given the piece-wise function (1) find a) f(-3) b) c) d) Solution: a) Note that to find , we need to use the middle function. That is, the function that is in the interval where -3 is included. Thus (2) b) Here, we use the middle function again because that is…
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Limits-One-Sided
Problem 96: Given the piece-wise function (1) find . Solution: Since this is a piece-wise function, we first need to find and to see if exists. That is, a) (2) b) (3) Since , (4)
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Solving Polynomials
Problem 95: Given the polynomial , a) solve for , and b) find the interval where the polynomial is nonzero. Solution: a) Let’s solve for . (1) b) That is, the polynomial is zero at . Therefore, the polynomial in nonzero at (2)
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Radical Equations
Problem 94: Solve the equation . Solution: We will solve for and then check our solution. (1) According to our steps above, the solution is given by . But is it? Let’s check. This is important to do. That is, if , then (2) Note that even after solving for , this equation…
One Problem at a Time
