The number 20! ÷ (10! × 10!) works out to a whole number. What is the highest power of 2 that divides it?

Count the twos in each factorial separately. In 20!: ⌊20/2⌋ + ⌊20/4⌋ + ⌊20/8⌋ + ⌊20/16⌋ = 10 + 5 + 2 + 1 = 18. In 10!: 5 + 2 + 1 = 8, and there are two copies of 10! underneath, so 16 twos cancel away. That leaves 18 − 16 = 2, i.e. $2^2$. Worked out in full, 20! ÷ (10! × 10!) $=184756=4\times 46189$, and 46189 is odd, which confirms the exponent is exactly 2. Subtracting only one 8 instead of two gives the wrong exponent 10.