Question

Find the number of n digit numbers formed using the first 5 natural numbers, which contain the digits \[2\& 4\] essentially.

Answer 1

Hint: We use permutation and combination formulas to find the number of n digits. Because a combination of 5 natural numbers is used to make n digit numbers. After the combination formula, we use permutation.
We use the first combination formula and then permutation formula to calculate required n digit numbers.

Complete step-by-step answer:
Given: first 5 natural numbers are as \[1,2,3,4,5\]
Here 2 and 4 are essentially digits to make n digit number. This implies that the minimum value of n is 2 and maximum digit is 5.
For example: \[24\] and \[42\] are two possible numbers made by 2 and 4.
For 3-digit numbers: we use permutation and combination formulas.
When 1 is chosen as $3^{rd}$ digit of number, we get
Numbers are possible in this case as \[3! = 3 \times 2 = 6\]
Similarly in case of digits \[3\& 5\]
Hence, numbers are possible are given as = \[6 + 6 + 6 = 18\]
For 4-digit numbers: From combination and permutation methods
We can find the combination of digits \[1,3,5{ = ^3}{C_2}\]
After combination, permutation can be given for all three combinations as \[ = 4! \times 3 = (4 \times 3 \times 2 \times 1) \times 3\]\[ = 72\]
Hence, total numbers having 4 digits \[ = 72\]
For 5-digit numbers: From combination and permutation methods
We can find the combination of digits \[1,3,5{ = ^3}{C_3}\]
After combination, permutation can be given for all three combinations as \[ = 5! \times 1\]\[ = \left( {5 \times 4 \times 3 \times 2 \times 1} \right) \times 1\]\[ = 120\]
Hence, total numbers having 5 digits\[ = 120\]
Hence, total numbers possible are given as
\[ = 2 + 18 + 72 + 120\]
\[ = 212\]

There are a total of 212 n digit numbers that can be formed.

Note: In mathematics, the method of arranging all the members of a set of data into some order is known as permutation. Permutations occur when different orderings on certain finite sets. The combination is defined as a way of selecting items. In combination, unlike permutations, the order of selection does not matter.