Question 890985
A binary number consists only 0s and 1s.
A) How may different seven-digit binary numbers are there?
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The first digit cannot be 0, and must be 1.
There is 1 way to choose the first digit.
There are 2 ways to choose the second digit, either 0 or 1.
There are 2 ways to choose the third digit, either 0 or 1.
There are 2 ways to choose the fourth digit, either 0 or 1.
There are 2 ways to choose the fifth digit, either 0 or 1.
There are 2 ways to choose the sixth digit, either 0 or 1.
There are 2 ways to choose the seventh digit, either 0 or 1.

That's 1󫎾󫎾󫎾 = 2<sup>6</sup> = 64 seven-digit numbers.
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B) How may different seven-digit binary numbers are there which begin with a 1
and which contain exactly four 1s.
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1 _ _ _ _ _ _

There are 6 positions to place the other 3 1's.
That's 6 positions choose 3 or C(6,3) = 20.  We
place 0's in the remaining 3 positions.

Answer: 20
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C) How may different seven-digit binary numbers are there which contain exactly
four 1s which are all adjacent(next to each other)?
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The first digit must be 1, and since all four 1's must be together, 
there is only 1,

1111000

Answer: 1

Edwin</pre>