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How many positive integers can be expressed as a product of two or more of the prime numbers 5, 7, 11, and 13 if no one product is to include the same prime factor more than once?
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A reading list for a humanities course consists of 10 books, of which 4 are biographies and the rest are novels. Each student is required to read a selection of 4 books form the list, including 2 or more biographies. How many selections of 4 books satisfy the requirements?
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There are 4 fiction books and 6 non-fiction ones, while 3 of the non-fiction are biographies. Now choose 3 books from the total 10 books. What is the probability of at least one fiction and no more than 1 biography are selected?
Give your answer as a fraction.
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To set a three-letter password for a website, a person selects two letters from the 26 alphabet and use one letter twice. How many different passwords are possible?
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In how many different ways can 3 boys and 3 girls be seated in a row of 6 chairs such that the girls are not separated, and the boys are not separated?
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In how many ways can three couples sit in a row such that each couple sit together?
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On a bookshelf, Pat arranges 7 different books: 2 history books, 3 philosophy books, and 2 science books. If Pat arranges the books so that the history books are next to each other, the science books are next to each other, and the philosophy books are next to each other, how many different arrangements are possible?
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Sid intended to type a seven-digit number, but the two "3" he meant to type did not appear. What appeared instead was the five-digit number 52115. How many different seven-digit numbers could Sid have meant to type?
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A fair coin is tossed 6 times. What is the probability of getting no any two heads on consecutive tosses?
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How many integers between 1 and $$10^{21}$$ are such that the sum of their digits is 2?
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Four different toys need to be assigned to three kids such that each kid receives at least one toy. In how many ways can these toys be distributed?
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Three typists work for four different companies. Each company has one document, and each document can be assigned to any of the three typists. What is the probability that three typists all receive documents?
Give your answer as a fraction.
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Point A (a, b) is in the xy-plane where both a and b are positive integers. What is the number of point A such that a+b < 200?
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How many points (r, s) can be formed so that r < s, and that the x and y-coordinates of the point are both selected from odd integers between 1 and 399, inclusive?
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Four guests A, B, C and D have to be assigned into three different rooms (one double room and two single rooms) such that A and B won`t have to stay in the double room at the same time. How many ways in total can they be assigned into these rooms?
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Four guests A, B, C and D have to be assigned into three different double rooms (at least one guest has to be assigned to each room) such that A and B won`t have to stay in the double room at the same time. How many ways in total can they be assigned into these rooms?
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Linda has 4 jackets, 3 pants and 2 pairs of shoes. She could choose 1 jacket, 1 pant and 1 pair of shoes every day. However, 1 jacket and 1 pant don`t match, so they cannot be selected together. In such circumstance, for how many days can she wears differently?
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Linda has 4 jackets, 3 pants, 2 pairs of shoes and 4 scarfs. She could choose 1 jacket, 1 pant and 1 pair of shoes every day. However, she could either wear 1 of the 4 scarfs or just wear no scarf. In such circumstance, for how many days can she wears differently?
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In how many different ways can 3 identical green shirts and 3 identical red shirts be distributed among 6 children such that each child receives a shirt?
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In how many ways can 4 kids (2 boys and 2 girls) be selected out of 5 couples of boy-girl twins such that no kids are selected from the same couple?
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