Help please (multiplication with fractions)

\(\frac{7}{6}\times \frac{8}{7} \times ...... \times \frac{40}{39}= \frac{40}{6}= \frac{20}{3}\)

The numerator of the n-th fractions just cancels out with the denominator of the (n+1)-th fractions

You use 7 through 40 divided by 6 through 39. The common factors are 7 through 39. You are left with merely 40/6 which equals 6 and 2/3.

7/6 x 8/7 x 9/8 x 10/9 x 11/10 x 12/11 x 13/12 x 14/13 x 15/14 x 16/15 x 17/16 x 18/17 x 19/18 x 20/19 x 21/20 x 22/21 x 23/22 x 24/23 x 25/24 x 26/25 x 27/26 x 28/27 x 29/28 x 30/29 x 31/30 x 32/31 x 33/32 x 34/33 x 35/34 x 36/35 x 37/36 x 38/37 x 39/38 x 40/39

The numerator is  40!/6!   and the denominator is  39!/5!

\(\frac{40!}{6!}\div \frac{39!}{5!}\\ =\frac{40!}{6!}\times \frac{5!}{39!}\\ =\frac{40}{6}\times \frac{1}{1}\\ =6\frac{4}{6}\\ =6\frac{2}{3}\\ \)