Don't use your calculator!

Since $10! = 10\times 9 \times 8 \times (7!),$

$\frac {6! \times 7!}{10!} = \frac {6! \times 7!}{10 \times 9 \times 8 \times 7!} = \frac {6 \times 5 \times 4 \times 3 \times 2 \times 1}{10 \times 9 \times 8}.$ Then, we can expand the denominator: $\frac {6 \times \boxed {5} \times \boxed {4} \times \boxed {3} \times \boxed {2} \times 1}{\underbrace{\boxed {5} \times \boxed {2}}_{10} \times \underbrace{\boxed {3} \times 3}_{9} \times \underbrace{\boxed {4} \times 2}_{8}},$ and canceling common terms gives: $\frac {6 \times 1}{3 \times 2} = \frac{6}{6}=1.$

I noticed that

$\huge6!=720 = 8 \times 9 \times 10=\dfrac{10!}{7!}$

$\huge\Rightarrow \dfrac{6!7!}{10!}=1$

This is a classic case of Catalan Number: knowing the first few terms of the sequence, we get $6 \times 7 = 42 = C_5 = \frac {1}{5+1} \cdot { 10 \choose 5} = \frac {1}{6} \cdot \frac {10}{5! \times 5!} = \frac {10!}{6! \times 5!} \Rightarrow 1 = \frac {10!}{6! \times 7!}$ . Reciprocate the expression yields the desired answer of $\boxed{1}$ .

6!7!/10!

=6!/(10.9.8) [divide top and bottom by 7!]

=6.5.4.3.2.1/10.9.8 [write 6! out]

=3.2.5.2.2.3.2/5.2.3.3.2.2.2 [write as product of primes]

=1

6 x 5 x 4 x 3 x 2 x 1 x 7 x 6 x 5 x 4 x 3 x 2 x 1 / 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 then cancel it all out

6! x 7!/10!

=6! x 7! /10 x 9 x 8 x7!

=6!/10 x 9 x 8

=720/720=1

6! x 7!/10!=6! x 7!/10 x 9 x 8 x7!

We omit 7! It is reduced to 6x5x4x3x2/3x3x2x4x2x5=1

6! X 7! / 10 x 9 x 8 x 7! factorial seven cancelled each other then 720 / 720 = 1

$6!×7!=2^2×3^2×(2^2)^2×5^2×(2×3)^2×7=2^8×3^4×5^2×7$

$10!=2×3×2^2×5×(2×3)×7×2^3×3^2×(2×5)=2^8×3^4×5^2×7$

$6!×7!=10!\Rightarrow\frac{6!×7!}{10!}=\boxed{1}$

(6! 7!)/10! =(1 2 3 4 5 6 1 2 3 4 5 6 7)/1 2 3 4 5 6 7 8 9 10 =(1 2 3 4 5 6)/(8 9*10) =720/720 =1

simple... $\frac{6!*7!}{10!}$ = $\frac{1*2*3*4*5*6}{8*9*10}$ =1 therefore, ans.= 1

6! X 7! /10! = 6!/(8 * 9 * 10) = 1 as both are 720 or by canceling like terms in 5 * 2, 3 * 2, 2 * 2 and 3

6!= 6 x 5 x 4 x 3 x 2 x 1

7!= 7 x 6 x 5 x 4 x 3 x 2 x 1

10!= 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 10 x 9 x 8 x 7!

6! x 7!/ 10! = 6! x 7!/10 x 9 x 8 x 7! = 6!/ 10 x 9 x 8 = 6 x 5 x 4 x 3 x 2 x 1/ 10 x 9 x 8 = 6 x 5 x 3 x (4 x 2 x 1)/10 x 9 x 8 =6 x 5 x 3 x 8/10 x 9 x 8 = 6 x 5 x 3/ 10 x 9 =90/90= 1

I included every step just in case anyone is confused! Hope it helps 😊

6! 7! / 10! = 6 5 4 3 2 1 7!/10 9 8 7! =2 3 5 4 3 2 1/2 5 3 3 4 2 =2 3 5 4 3 2 1/2 3 5 4 3 2 =1

1x2x3x4x5x6x1x2x3x4x5x6x7/1x2x3x4x5x6x7x8x9x10 =2x3x4x5x6/8x9x10=1

7!×6!\10!
6!/10×9×8

720/720

1

Easy one but AWESOME

$\frac{6! * 7!}{10!} = \frac{6! * 7!}{10*9*8 * 7!} = \frac{6*5*4*3*2}{10*9*8}$ $6!: 30, 120, 360, 720$ $10*9*8: 90, 720$ $\frac{720}{720} = 1$

simple fac logic 6! 7!/10! 6!/10 9*8 6!=720 so answer 1

6! x 7! = 3628800 10! = 3628800 3628800/3628800=1

Really easy!

6! 7!/10!=6! 7!/10 9 8*7!=720/720=1

(6!X7!)/10! =(720X5040)/3628800 =3628800/3628800 =1

(6!×7!)/10! (6×5×4×3×2×1×7!)/(10×9×8×7!)=(6×5×4×3×2×1)/(10×9×8)=1

7!6!/10! 7!6!/10 9 8 7! 6!/10 9 8 120 6/720 720/720 1

In easier form, 6 X 5 X 4 X 3 X 2 X 1 X 7 X 6 X 5 X 4 X 3 X 2 X 1

        10 X 9 X 8 X 7 X 6 X 5 X 4 X 3 X 2 X 1

= 6 X 5 X 4 X 3 X 2 X 1 _ _ _ _ _ __ 10 X 9 X 8 = 1

Just expand this and cancel all the factors and rada u get the answer