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Combinations

If a coin is flipped twenty times, how many ways are there to get exactly ten tails and ten heads?

With Restriction

Suppose \(5\) distinct numbers are chosen from nine integers \(1, 2, \ldots, 9\) to create a \(5\)-digit number. Given that the digits chosen must consist of \(3\) odd numbers and \(2\) even numbers, how many distinct \(5\)-digit numbers can be created?

3 red balls and \(12\) blue balls are randomly placed in a row. What is the probability that no two red balls are adjacent?

Suppose a \(4\)-digit number is created from \(4\) distinct integers chosen among \(\{1, 2, 3, 4, 5, 7, 9\}.\) How many such \(4\)-digit numbers include both \(1\) and \(2\) but neither begin nor end with \(1?\)

In Sue's Fruit & Flowers Market, there are \(8\) different kinds of fruit, including peaches and apples. There are also \(6\) different kinds of flowers, including roses and lilies. How many ways are there to make a gift basket such that the basket has \(3\) different kinds of fruit including peaches and \(2\) different kinds of flowers with no roses?

From a group of \(11\) students including \(A\) and \(B\), \(4\) people are selected to be arranged in a line. If both \(A\) and \(B\) must be among the \(4\) selected students, how many distinct line-ups are possible?

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