Maths-
General
Easy

Question

The number of different seven digit numbers that can be written using only the three digits 1, 2 and 3 with the condition that the digit 2 occurs twice in each number is-

  1. 7P2 . 25    
  2. 7C2 . 25    
  3. 7C2 . 52    
  4. None of these    

Hint:

Here, we will find the number of ways of the arrangement of the digit if two occur exactly twice in each seven-digit number by using the combination formula. Then by using the powers, we will find the number of ways of arranging the remaining five-digit number. We will then multiply the number of the arrangement of digits in both the cases to get the number of different seven-digit numbers.

Formula Used:
We will use the following formula:
1) Combination is given by the formula 
C presuperscript n subscript r space equals space begin inline style fraction numerator n factorial over denominator r factorial space left parenthesis n minus r right parenthesis factorial end fraction end style end subscript

2) Factorial is given by the formula

n! = ncross times(n-1)! 

The correct answer is: 7C2 . 25


    Detailed Solution
    We are given the number of different seven-digit numbers that can be written using only three digits 1,2 and 3. Therefore,
    T o t a l space n u m b e r space o f space D i g i t s space equals 7
    We are given that the digit two occurs exactly twice in each number.
    Thus, the digit two occurs twice in the seven digit number.
    Now, we will find the number of ways of arrangement of the digit two in the seven digit number by using combination.


    T o t a l space n u m b e r space o f space w a y s space t h a t space t h e space d i g i t space t w o space o c c u r s space e x a c t l y space t w i c e space i n space e a c h space n u m b e r space equals scriptbase C subscript 2 end scriptbase presuperscript 7
    N o w comma space t h e space r e m a i n i n g space f i v e space d i g i t s space c a n space b e space w r i t t e n space u sin g space t w o space d i g i t s space 1 space a n d space 3 space i n space 2 to the power of 5 space space w a y s.

    We will now find the total number of seven digit number by multiplying the number of ways of arrangement in both the cases. Therefore
    T o t a l space n u m b e r space o f space s e v e n space d i g i t space n u m b e r space equals scriptbase C subscript 2 end scriptbase presuperscript 7 cross times space 2 to the power of 5
    N o w space b y space u sin g space t h e space f o r m u l a space scriptbase C subscript r end scriptbase presuperscript n equals fraction numerator n factorial over denominator left parenthesis n minus r right parenthesis factorial r factorial space end fraction space comma space w e space g e t
    rightwards double arrow space space T o t a l space n u m b e r space o f space s e v e n space d i g i t space n u m b e r space equals fraction numerator 7 factorial over denominator left parenthesis 7 minus 2 right parenthesis factorial 2 factorial end fraction cross times 2 to the power of 5


    W e space k n o w space t h a t space t h e space f a c t o r i a l space c a n space b e space w r i t t e n space b y space t h e space f o r m u l a space n factorial equals n cross times left parenthesis n minus 1 right parenthesis factorial space space comma space s o space w e space g e t

    rightwards double arrow space space T o t a l space n u m b e r space o f space s e v e n space d i g i t space n u m b e r space equals fraction numerator 7 cross times 6 cross times 5 factorial over denominator 5 factorial 2 factorial end fraction cross times 2 to the power of 5

    rightwards double arrow space space T o t a l space n u m b e r space o f space s e v e n space d i g i t space n u m b e r space equals fraction numerator 7 cross times 6 over denominator 2 factorial end fraction cross times 2 to the power of 5

    Simplifying the expression, we get

    rightwards double arrow space space T o t a l space n u m b e r space o f space s e v e n space d i g i t space n u m b e r space equals 7 cross times 6 cross times 2 to the power of 4

    Multiplying the terms, we get
    rightwards double arrow space space T o t a l space n u m b e r space o f space s e v e n space d i g i t space n u m b e r space equals 672
    Therefore, the number of different seven-digit numbers that can be written using only three digits 1,2 and 3 is 672.

    We know that there is not much difference between permutation and combination. Permutation is the way or method of arranging numbers from a given set of numbers such that the order of arrangement matters. Whereas combination is the way of selecting items from a given set of items where order of selection doesn’t matter. Both the word combination and permutation is the way of arrangement. Here, we will not use permutation because the order of toys is not necessary.

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