Question

Find the value of \[\dfrac{{7!}}{{5!}}\]?

Answer 1

Hint: Here in this question, they have used the \[ \div \] operator. This operator does the division and we are dividing the two terms. In this question we also see ! symbol this represents the factorial, the factorial is defined as \[n! = n \times (n - 1) \times (n - 2) \times ... \times 2 \times 1\].

Complete step-by-step answer:
Here in this question, they have used a multiplication symbol. We have mathematical operations namely addition, subtraction, multiplication and division. + represents the addition, - represents subtraction, \[ \times \]represents the multiplication and \[ \div \]represents the division. The multiplication is a repeated addition and the division is the repeated subtraction.
Now consider the given question
\[ = \dfrac{{7!}}{{5!}}\]
By the definition of factorial the above term can be written as
\[ = \dfrac{{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{5 \times 4 \times 3 \times 2 \times 1}}\]
Now we will cancel the terms which are present commonly in both the numerator and the denominator. On cancelling we get
\[ = 7 \times 6\]
On multiplying the 7 and 6 we get
\[ = 42\]
This can also be solved by other methods.
consider
\[ = \dfrac{{7!}}{{5!}}\]
This can be written as
\[ = \dfrac{{7 \times 6 \times 5!}}{{5!}}\]
The \[5!\] will get cancels
\[ = 7 \times 6\]
On multiplying the 7 and 6 we get
\[ = 42\]
Hence the value of \[\dfrac{{7!}}{{5!}}\] is 42.
Since the given question is in the form of fraction we have not done division. Sometimes the terms or numbers which are present in the numerator and denominator will get cancelled.
So, the correct answer is “42”.

Note: To solve the mathematical problems we have mathematical operations. There are for mathematical operations namely addition, subtraction, multiplication and division. + represents the addition, - represents subtraction, \[ \times \]represents the multiplication and \[ \div \]represents the division. The multiplication is the repeated addition. ! indicates the factorial. In general the factorial is defined as \[n! = n \times (n - 1) \times (n - 2) \times ... \times 2 \times 1\] where n is a positive integer.