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IRISSAK [1]
3 years ago
11

How many integers from 1 through 100,000 contain the digit 6 exactly once?

Mathematics
1 answer:
wariber [46]3 years ago
4 0
Consider the set of all (not-all-zero) decimal strings of length 6. This is the set of strings


000001
000002
...
099998
099999
100000

There are obviously 100,000 strings in this set, so we have a one-to-one correspondence to the integers between 1 and 100,000. Think of any string starting with 0s as the number with the leading 0s chopped off.

There are two choices for the first digit, either 0 or 1, but a number can only contain a 6 if the first digit is 0; otherwise, the number would exceed 100,000. For every digits place afterward, if a given digits place contains a 6, then the remaining four places have 9 possible choices each, choosing from 0-9 excluding 6. If we fix the 6 in, say, the second digits place, then the number of integers between 1 and 100,000 containing exactly one 6 is


1\cdot1\cdot9^4=6561


where the first 1 refers to the only choice of 0 in the first digits place, the second 1 refers to the unique 6 in the next place, and the remaining four places are filled with one of 9 possible choices.


Now, notice that we can permute the digits of such a number in 5 possible ways. That is, there are 5 choices for the placement of the 6 in the number, so we multiply this count by 5.

5(1\cdot1\cdot9^4)=32,805
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in triangle WXY, if the measure of angle w is five less than twice the measure of angle y and the measure of angle x is 21 more
Vikki [24]

Considering that the sum of the internal angles of a triangle is of 180º, the measures of the angles in triangle WXY are given as follows:

x = 62º, y = 41º, w = 76º.

<h3>What is the sum of the internal angles of a triangle?</h3>

The sum of the internal angles of a triangle is of 180º. In this problem, the angles are w, x and y, hence:

x + y + w = 180.

The measure of angle w is five less than twice the measure of angle y, hence:

w = 2y - 5.

The measure of angle x is 21 more than the measure of angle y, hence:

x = y + 21.

Replacing in the first equation, we can find angle y as follows:

y + 21 + y + 2y - 5 = 180.

4y = 164.

y = 164/4

y = 41.

Then:

  • w = 2y - 5 = 2(41) - 5 = 76.
  • x = y + 21 = 41 + 21 = 62.

The measures of the angles are given as follows:

x = 62º, y = 41º, w = 76º.

More can be learned about the sum of the internal angles of a triangle at brainly.com/question/25215131

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