Find the number of 4 -digit numbers that can be formed using...

Key Concepts

Even Number (Parity) Constraints

In number theory, an even number is one that is divisible by 2, meaning its last digit must be even. When counting number arrangements that must be even, the restriction applies to the unit's place, reducing the pool of available digits for that position and affecting the choices available for the remaining positions.

Positional Restrictions

Positional restrictions occur when specific conditions are applied to individual places or positions within an arrangement. For example, in forming numbers, constraints such as not repeating digits or specifying that a particular digit must appear in a given position are common restrictions.

Permutations

Permutations refer to the different ways in which a set of items can be arranged, where the order of arrangement matters. In problems where repetition is not allowed, the number of permutations is determined by counting how many choices are available for each position and multiplying these choices together.

Multiplication Principle

The multiplication principle is a fundamental counting technique that states that if an event can occur in 'm' ways and another event can occur independently in 'n' ways, then the two events together can occur in m × n ways. This principle is used to determine the total number of outcomes when filling different positions sequentially.

Transcript

00:01 So in the given question we are told to find the number of four digit numbers that can be formed.

00:06 The number of four digit numbers that can be formed using the digits.

00:16 1, 2, 3, 4 and 5.

00:23 And we are told that there is no repetition.

00:26 That is no digits are repeated.

00:31 So and after finding the number of four digit numbers that can be formed without repeating the digits with using the digits 1 to 5 we are also asked how many of these numbers are even.

00:46 So first in order to find the number of 4 digit numbers using the digits numbers from 1 to 5 we can draw 4 boxes over here which would represent the four places of a four digit number and now when we are taking writing when we are forming a number let's start from the thousands place since there is no repetition we have first we have five choices out of which we can select one number and place it in the thousand's place next since we have used up one of our choices we have only four left to place in the hundreds place then there is only three for the next place and two for the next so the number of four digit numbers can be found by multiplying these numbers five times four times three times two which is equal to twenty times six that is equal to one hundred number and now we are asked how many of these numbers are even right so to find that how many of these numbers are even we can draw another box with four spaces and over here what we can do is let's say for a number to be even it should have either two or four in its once place, right? so there should be two or four, that is we have two choices out of which we can place one in the once place...

Please give Ace some feedback