Basics of number theory

Number Theory Level 3

Suppose there are $x$ ways to arrange the elements of $99$ element set. Find the number of digits of $x$

The answer is 156.

1 solution

Ashish Menon
Mar 6, 2016

The answer is $P (99)$ which has $156$ digits.

@Ashish Siva How do you know $P(99)$ has $156$ digits?

Rushikesh Jogdand - 5 years, 3 months ago

I found the answer and put it in th place where i have to submit the answer. It allows only 33 digits and it shows that it has 156 digits, so i got the answer :P

Ashish Menon - 5 years, 3 months ago

I was asking for a general method.

Rushikesh Jogdand - 5 years, 3 months ago

@Rushikesh Jogdand – If not what i did was i grouped in channels of 3 and counted the no. of groups and multiplied it with 3.

Ashish Menon - 5 years, 3 months ago

@Ashish Menon – Please explain.

Rushikesh Jogdand - 5 years, 3 months ago

@Rushikesh Jogdand – It is easier to count 3 numbers at a time than to count 1 number at a time. For example in the number 123456, it is difficult to count one by one that there are 6 digits but if 1 group them into 3 numbers like 123|456, then it is easier to count, there are 2 groups . So, the number of digits are 2×3=6. Do the same thing for the answer

Ashish Menon - 5 years, 3 months ago

@Ashish Menon – But how to calculate $99!$

Rushikesh Jogdand - 5 years, 3 months ago

@Rushikesh Jogdand – 1) Create an array ‘res[]’ of MAX size where MAX is number of maximum digits in output.
2) Initialize value stored in ‘res[]’ as 1 and initialize ‘res size’ (size of ‘res[]’) as 1.
3) Do following for all numbers from x = 2 to n.
……a) Multiply x with res[] and update res[] and res

size to store the multiplication result.

Now, how to multiply (res[],x)

1) Initialize carry as 0.
2) Do following for i = 0 to res size – 1.
….a) Find value of res[i] * x + carry. Let this value be prod.
….b) Update res[i] by storing last digit of prod in it.
….c) Update carry by storing remaining digits in carry.
3) Put all digits of carry in res[] and increase res size by number of digits in carry.

If you still dont understand, i can give you an example.

Ashish Menon - 5 years, 3 months ago

@Ashish Menon – Although I didn't understand it, here's what I did

We have to find number of digits in $99!$ (in base $10$ , of-course)

So, we've to find out answer to $\log_{10} 99! +1$

$\log_{10}99!=\log_{10}(1\times 2\times 3\times \dots \times 99)=\log_{10}1 + \log_{10}2 + \log_{10}3 + \dots +\log_{10}99$

Any good calculator can find this

I would like to know what you did in details.

Rushikesh Jogdand - 5 years, 3 months ago

Basics of number theory

The answer is 156.

1 solution

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