Can u please help me with 57 If the threedigit numbers are f

If the three-digit numbers are formed at random, calculate the probability that such a number will be: As described in (ii). As described in (iii). On a straight line, there are n spots to be filled in by either a dot or a dash. What is the number of the distinct groups of resulting symbols? What is this number if n = 5, 10, 15, 20, and 25? A childs set of blocks consists of two red, four blue, and five yellow blocks. The blocks can be distinguished only by color. If the child lines up the blocks in a row at random, calculate the following probabilities: Red blocks appear at both ends. All yellow blocks are adjacent. Blue blocks appear at both ends. Suppose that the letters C, E, F, F, l, and O are written on six chips and place into a box. Then the six chips are mixed and drawn one by one without replacement. What is the probability that the word OFFICE is formed?

Solution

5.7

i)

There are 11!/(2!4!5!) = 6930 ways to line them up.

If reds are at the ends, the we can only fix the middle in 9!/(4!5!) = 126 ways.

Thus,

P = 126/6930 = 0.018181818 [answer]

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ii)

If all yellow blocks are adjacent, it\'s just like they are just one body.

Thus, it\'s like there are just 7 books. Thus, there are 7!/(2!4!1!) = 105 ways to do this.

Thus,

P = 105/6930 = 0.015151515 [answer]

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iii)

If two of the blue blocks are already fixed, then only 2 blue blocks are free to be ordered in the middle.

There are 9!/(2!2!5!) = 756 ways.

Thus,

P = 756/6930 = 0.109090909 [answer]