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How many six-letter words can one generate with the letters of the word CANADA?

  • Last Updated : 28 Nov, 2021

In mathematics, permutation is known as the process of arranging a set in which all the members of a set are arranged into some series or order. The process of permuting is known as the rearranging of its components if the set is already arranged. Permutations take place, in more or less important ways, in almost every area of mathematics. They frequently appear when different commands on certain finite sets are considered.

Permutation Formula

In permutation r things are selected from a set of n things without any replacement. In this order of selection matter.

nPr = (n!)/(n – r)!

where

n = set size, the total number of items in the set

r = subset size, the number of items to be selected from the set

Combination

A combination is an act of choosing items from a group, such that (not like permutation) the order of choice does not matter. In smaller cases, it is possible to count the number of combinations. Combination refers to the union of n things taken k at a time without repetition  In combination you can select the items in any order. To those combinations in which re-occurrence is allowed, the terms k-selection or k-combination with replication are frequently used.

Combination Formula

In combination r things are selected from a set of n things and where the order of selection does not matter

nCr = n!⁄((n-r)! r!)

Here, 



n = Number of items in set

r = Number of items selected from the set

How many six-letter words can one generate with the letters of the word CANADA?

Solution:

Case 1: Where one A can’t be right before or after another A (like AA or AAA)

There are six positions, numbered 1 through 6, to be assigned to the six letters.

The positions assigned to the 3 A’s may be (1,3,5) i.e., A*A*A, (1,3,6) i.e., A*A**A,

(1,4,6) i.e., A**A*A or (2,4,6) i.e., *A*A*A the asterisks must be substituted with

letters C, N, D in order to abide by the rules. So there are only 4 ways to assign

positions to the 3 A’s.

The remaining 3 letters are distinct, so they can be placed in 3! = 6 different ways.



Therefore, the number of words you can make using 6 letters from “CANADA” only once

where one A can’t be right before or after another A (like AA or AAA) is

4*6 = 24.

Case 2: Where one A can be right before or after another A (like AA or AAA)

The word ‘CANADA’ contains 3 A’s, 1 C, 1 N, and 1 D.

Number of permutations of the letters of the given word = 6!/3! = 120.

Similar Questions

Question 1: How many five-letter words can one generate with the letters of the word India?

Solution:

Case 1:- where one I can’t be right before or after another I (like II)

There are five positions, numbered 1 through 5, to be assigned to the five letters.

The positions assigned to the 2 I’s may be (1,3) i.e., I*I**, (3,5) i.e., **I*I or

(2,4,) i.e., *I*I* the asterisks must be substituted with letters N,D,A in order to

abide by the rules. So there are only 3 ways to assign positions to the 2 I’s.

The remaining 3 letters are distinct, so they can be placed in 3! = 6 different ways.

Therefore, the number of words you can make using 5 letters from “INDIA” only once

where one I can’t be right before or after another I (like II) is

3*6 = 18.

Case 2 :- where one I can be right before or after another I (like II)

There are 60 different ways to arrange the 5 letters in “INDIA”.

Explanation:



The word ‘INDIA’ contains 2 I’s, 1 A, 1 N and 1 D.

Number of permutations of the letters of the given word =5!⁄2!=60.

Question 2: How many Seven-letter words can one generate with the letters of the word America?

Solution:

Case 1:- where one A can’t be right before or after another A (like AA)

There are Seven positions, numbered 1 through 7, to be assigned to the seven letters.

The positions assigned to the 2 A’s may be (1,3) i.e., A*A**, (3,5) i.e., **A*A,

(2,4,) i.e., *I*I* (4,6) i.e., ***A*A* (5,7) i.e., ****A*A the asterisks must be

substituted with letters M,E,R,I,C in order to abide by the rules. So there are only

5 ways to assign positions to the 2 A’s.

The remaining 5 letters are distinct, so they can be placed in 5! = 120different ways.

Therefore, the number of words you can make using 7 letters from “AMERICA” only once

where one A can’t be right before or after another A (like AA) is

5×120 = 600.

Case 2 :- where one A can be right before or after another A (like AA)

There are 2520 different ways to arrange the 7 letters in “AMERICA”.

Explanation:

The word ‘AMERICA’ contains 2 A’s, 1 M, 1 E, 1R, 1I and 1C.

Number of permutations of the letters of the given word =7!⁄2!=2520.

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