Question:
The total number of $4$-digit numbers in which the digits are in descending order, is
The total number of $4$-digit numbers in which the digits are in descending order, is
Updated On: Sep 3, 2024
- $^{10}C_4 \times 4!$
- $^{10}C_4 $
- $\frac{10!}{4!}$
- None of these
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The Correct Option is B
Solution and Explanation
Total number of arrangements of $10$ digits $0,1,2, \ldots ., 9$
by taking 4 at a time $={ }^{10} C_{4} \times 4 !$
We observe that in every arrangement of $4$ selected digits there is just one arrangement in which the digits are in descending order.
$\therefore$ Required number of $4$-digit numbers.
$=\frac{{ }^{10} C_{4} \times 4 !}{4 !}={ }^{10} C_{4}$
by taking 4 at a time $={ }^{10} C_{4} \times 4 !$
We observe that in every arrangement of $4$ selected digits there is just one arrangement in which the digits are in descending order.
$\therefore$ Required number of $4$-digit numbers.
$=\frac{{ }^{10} C_{4} \times 4 !}{4 !}={ }^{10} C_{4}$
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Concepts Used:
Combinations
The method of forming subsets by selecting data from a larger set in a way that the selection order does not matter is called the combination.
- It means the combination of about ‘n’ things taken ‘k’ at a time without any repetition.
- The combination is used for a group of data where the order of data does not matter.
- For example, Imagine you go to a restaurant and order some soup.
- Five toppings can complement the soup, namely:
- croutons,
- orange zest,
- grated cheese,
- chopped herbs,
- fried noodles.
But you are only allowed to pick three.
- There can be several ways in which you can enhance your soup with savory.
- The selection of three toppings (subset) from the five toppings (larger set) is called a combination.
Use of Combinations:
It is used for a group of data (where the order of data doesn’t matter).