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The answer works out to \(\dfrac{7 \pi}{32}\)

113 , 133 , 153 , 173 , 193 , 213 , 233 , 253 , 273 , 293 , 313 , 333 , 353 , 373 , 393 , 413 , 433 , 453 , 473 , 493 , 513 , 533 , 553 , 573 , 593 , 613 , 633 , 653 , 673 , 693 , 713 , 733 , 753 , 773 , 793 , 813 , 833 , 853 , 873 , 893 , 913 , 933 , 953 , 973 , 993>> Total of all 3-digit numbers =  45

Thank you! That makes sense

Edit: I also noticed that point S is also a terminal point of \(\theta \)/2\(\)

find at what value of the variable x the expressions -70x-1x and 139x+70 are equal

Equate the 2 expressions:

- 70x - 1x = 139x + 70

 - 71x =139x + 70

- 71x - 139x = 70

- 210x = 70

x = 70 / - 210

x = - 1 / 3 

N mod 3 =2, 

N mod 5 =2,   

N mod 7 =2, solve for N

N = 105C + 2, where C= 0, 1, 2, 3,.......etc.

When C = 1, then:

N=[105 * 1  +  2] = 107 - number of students in RSM online classes.

Check: 107 is 1 less than a multiple of 3, or: 107 + 1 =108 - which is a multiple of 3

              107 is 3 less than a multiple of 5, or: 107 + 3 =110 - which is a multiple of 5

              107 is 5 less than a mulriple of 7, or: 107 + 5 =112 - which is a multiple of 7 

The probability of winning a super prize is the sum of the probabilities of matching at least two white balls and the probability of matching the red SuperBall.

The probability of matching at least two white balls is 1 minus the probability of not matching any white balls. To match none of the white balls, we first choose three distinct numbers from 1 to 12 that do not match the three white balls that are drawn. This can be done in 12 choose 3 ways (or 220 ways) for the white balls, and then in 8 choose 1 ways for the red SuperBall, for a total of 220*8 = 1760 ways. Since there are 12 choose 3 ways to choose 3 numbers from 1 to 12, the probability of not matching any white balls is 1760/220 = 8/11. Therefore, the probability of matching at least two white balls is 1 - 8/11 = 3/11.

The probability of matching the red SuperBall is 1/8, since there are 8 red SuperBalls out of a total of 20.

Therefore, the probability of winning a super prize is (3/11) + (1/8) = 29/44.

See https://web2.0calc.com/questions/algebra_73437

See https://web2.0calc.com/questions/algebra_73437

Fix one of the scientists at the head of table:

The other 4  can sit either to the left or to the right, or split 2 to the left and 2 to right, or 1 to right and 3 to the left or vise versa (it makes no difference) in this many ways:

4! = 24 ways

The remaining people (2 + 5 + 2 + 3 = 12) can be seated in:

12! ==479,001,600 ways - since there is no restriction to their seating.

So, the total number of ways =24 x 479,001,600 = 11,496,038,400 

See https://web2.0calc.com/questions/algebra_73437

See https://web2.0calc.com/questions/algebra_73437

See https://web2.0calc.com/questions/algebra_73437

The smallest prime when divided by 60 leaves a remainder that is a composite number is:

109 mod 60 ==49 - which is a composite number ==7 x 7 ==49

The answer is 9.

The answer is 92.

The answer is 53.

Use substitution to get:

a≈1.21928  and b≈2.78072                     [Real
a≈2.78072  and b≈1.21928                     [Real

a≈2 - 4.9608 i  and  b≈2 + 4.9608 i        [Imaginary
a≈2 + 4.9608 i  and  b≈2 - 4.9608 i        [Imaginary

As follows:

See https://web2.0calc.com/questions/let-be-an-angle-such-that-p-is-the-terminal-point-of_1#5

The angle \(\theta\) goes anti-clockwise from the positive x-axis around to the red line.  The angle \(\theta/2\) would go half-way (again in an anti-clockwise direction) from the positive x-axis towards the red line.  It would therefore get to point Q.

is there an additional diagram or graph involved with this problem?