Question

Give five examples where the number of things counted is greater than a six-digit number.

Answer 1

- Hint: Find the value of n when the number of digits in n! exceeds 6 digits. Use the fact that the arrangement of n different things taken all at a time is given by n!. Hence generate five examples where the number of things counted would be more than 6-digit numbers.

Complete step-by-step solution -

For the number of digits in n! to be more than 6, we can select the value of n such that the power of 5 in n! is more than 6.
So, if we take n = 30, we have the power of 5 in 30! equals $\left[ \dfrac{30}{5} \right]+\left[ \dfrac{30}{25} \right]=6+1=7$, and hence 30! will have more than six digits.
Hence the five different examples where the number of things counted will be more than six-digit number are:
[i] The number of ways in which 30 persons can be seated in a line. This is equal to 30!
[ii] The number of 30 letter long words that can be formed using all the 30 letters of an alphabet. This is equal to 30!
[iii] The number of paths between two vertices in a complete graph of 30 vertices $\left( {{K}_{30}} \right)$. This is equal to 30!.
[iv]The number of subsets of a set of cardinality 30 ( $\because {{2}^{30}}>999999$ ).
[v] The number of relations from A to B where n(A) = 5 and n(B) = 6. This is equal to ${{2}^{5\times 6}}={{2}^{30}}>999999$

Note: Alternatively, we can think of examples from nature where the number of things to be counted is greater than a six-digit number like:
[i] The number of stars in the universe. The estimated number is about 1 billion trillion stars.
[ii] The number of atoms in 1 mole. This is equal to Avagadro's number which is approximately equal to $6.022\times {{10}^{23}}$
[iii] The number of microorganisms in a square metre of soil. There are approximately ${{10}^{9}}$ bacteria, ${{10}^{9}}$ protozoa, etc. in a square metre of soil.
[iv] Number of species of various organisms. The estimated number is about 1 trillion.
[v] Number of cells in the human body. The estimated number is about 37 trillion in a full-grown human body.