The number of 4 digit numbers without repetition that can be formed using the digits 1, 2, 3, 4, 5, 6, 7 in which each number has two odd digits and two even digits is

Concept:

Product rule:

If we can select object A in m ways and object B in n ways then we can select the object A and B in mn ways.

Calculation:

We have to form 4 digit numbers so we can do it in 4! ways.

There are 4 odd numbers and we have to select 2 odd numbers out of those which can be done in \(\rm ^4C_2\) ways.

We will calculate this as follows:

\(^4C_2=6\)

Similarly, there are 3 even numbers and we have to select 2 out of those which can be done in \(\rm ^3C_2\) ways.

Which is same as:

\(^3C_2=3\)

Therefore, using the product rule we can calculate the required number of formations as follows:

\(\rm ^4C_2\times ^3C_2\times 4! = 6\times 3\times 24\\ = 432 \)

The number of 4 digit numbers without repetition that can be formed using the digits 1, 2, 3, 4, 5, 6, 7 in which each number has two odd digits and two even digits is 432.