The highest power of 18 contained in 50C25
Answers
Answer:
Step-by-step explanation:
(5025)
=>50!25!∗(50–25)!
=>50!25!∗25!
Now let me inform one very common thing :Highest power of a prime number “p” in n!can easily be obtained by the simple formula
N=[np]+[np2]+[np3]+[np4]+..
*Note that here [x][1] denotes the highest/greatest integer not greater than x.
In numerator of 50!25!∗25! is N=50! and denominator is D=25!∗25!
Power of 3 in N=
[503]+[5032]+[5033]+[5034]+...=16+5+1+0+...=22
Power of 3 in D=
[253]+[2532]+[2533]+[2534]+...∗2=(8+2+0+...)∗2=20
similarly power of 2 in N=
[502]+[5022]+[5023]+[5024]+...=25+12+6+3+1+0+..=47
Power of 2 in D=
[252]+[2522]+[2523]+[2524]+...∗2=(12+6+3+1+0+...)∗2=44
So in 50!25!∗25! power of 3 is (22–20)=2
and in 50!25!∗25! power of 2 is (47-44)=3
so in order to get the highest power of 18=2∗32 that divide 50!25!∗25! is given by:min{[power−of−32],[power−of−21]}=min{[22],[31]}=min(1,3)=1
Thank you .. Suggest if edits are needed..
Answer:
The highest power of Any prime number(a) in a factorial (n!) Is given by [n/a] + [n/a²] + [n/a³] and so on where [] is the greatest integer function. So as soon as a^k becomes greater than n every term will be 0.
Now 50C25 is 50!/(25!25!). 18 is 3².2. 3² is limiting. If we find the highest power of 3 and divide it by 2, we'll get the highest power of 9 which will be the same as that of 18. Highest power of 3 in 50! Is [50/3] + [50/9]… = 22. So the highest power of 9 is 11.
Similarly for 25, the highest power of 3 is 10 => the highest power of 9 is 5. Since it's squared, the power of 9 becomes 10. So on dividing 9^11/9^10, we'll get one 9 left which equivalent to one 18 left (because 2s are in abundance). So, the highest power of 18 in 50C25 is 1.