Find the number of zeroes at the end of (51! -50!)

14 zeroes

Step-by-step explanation:

By legendre's factorial formula (which is quite intuitive if you take time to think about it), the number of zeroes in is zeroes.

Notice that 50 = 5^2 * 2, so it will contribute 2 more zeroes to the number.

Therefore, there will be 12+2 = 14 zeroes in total.

Rough explanation of why the above works:

Note that multiplying anything by 10 adds one zero at the end of a number, and 10=2*5.

In the prime factorization of a factorial function, the power of 5 will obviously be way smaller than the power of 2 (you could calculate exactly but i'm not going to do that here).

In this case, we need to find the power of 5's in the prime factorization of 50! * 50, which was demonstrated above.

You could find the power of 2's to absolutely confirm that it will work, but if you want to save time in a competition that will probably be irrelevant.

given that triangle abc is transformed two times to get triangle

a"b"c"

first abc is transformed in to a'b'c' as follows:

the coordinates of a are (1,-1) and transformed into a'(-1,1)

similarly b (4,-2) became b'(-4,2) and

c(7,-2) became c'(-7,2)

i.e. (x,y) becomes (-x,-y)

this is nothing but reflection about a point here origin.

thus first transformation is reflection on the origin.

next is exactly shifting 3 units down

source:

13

step-by-step explanation:

divide the cost of the total of sodas you bought by the cost of one soda, for example:

$24.57 divided by $1.89 is 13 which means you bought 13 sodas