How many ways can 5 children be arranged in a line such that I two of them Ram and Shyam are always together ii two of them Ram and Shyam are never together?

Solutions

(1)

Ram R Shyam S

No. of ways of arranging in which R & S together =4!×2!=48

No. of ways of arranging in which R & S not together = Total ways to arrange No. of ways of arranging in which R & S together.

=5!4!×2!

=12048=72.

How many ways can 5 children be arranged in a line such that I two of them Ram and Shyam are always together ii two of them Ram and Shyam are never together?
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How many ways can 5 children be arranged in a line such that I two of them Ram and Shyam are always together ii two of them Ram and Shyam are never together?

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1. Factorial:

(i) The continued product of first n natural numbers is called the " n factorial" and is denoted by n or n!.

(ii) n!=1×2×3×4×...×(n-1)×n

(iii) Factorials of proper fractions and negative integers are not defined.

(iv) 0!=1

(v) (2n)!n!=1⋅3⋅5....2n-12n

2. Fundamental Principle of Multiplication: 

If there are two jobs such that one of them can be completed in m ways, and when it has been completed in any one of these m ways, second job can be completed in n ways, then the two jobs in succession can be completed in m×n ways.

3. Fundamental Principle of Addition:

If there are two jobs such that they can be performed independently in m and n ways respectively, then either of the two jobs can be performed in m+ n ways.

4. Permutations/Arrangements:

(i) If n is a natural number and r is a positive integer such that 0≤r≤n, then  nPr=n!(n-r)!.

(ii) Permutation of n distinct objects taken r at a time=Prn.

(iii) Permutation of n distinct objects taken all at a time=Pnn=n!.

(iv) Permutation of n objects taken all at a time of which p are alike=n!p!.

(v) Permutations of n objects of which p are alike of one kind, q are alike of second kind and remaining all are distinct=n!p!  q!.

(vi) Permutation of n objects such that p objects are never together=n!-n-p+1!.

(vii) Permutation of n objects such that p objects are always together=n-p+1!.

(viii) Gap method:

Arrangement of boys and girls such that no two boys are together can be found by arranging all the girls first and then arranging the boys in the gaps.

(ix) Permutation of n objects taken r at a time when repetition is allowed=nr.

(x) Permutation of n objects taken all at a time when repetition is allowed=nn.

5. Circular Permutations:

(i) Instead of arranging the objects in a line, if we arrange them in the form of a circle, we call them circular permutations.

(ii) In circular permutations, what really matters is the position of an object relative to the others.

There are two types of circular permutations: (i) The circular permutations in which clockwise and the anticlockwise arrangements give rise to different permutations, e.g., seating arrangements of persons round a table.

(ii) The circular permutations in which clockwise and the anticlockwise arrangements give rise to same permutations. e.g., arranging some beads to form a necklace.

6. Important Points to remember:

(i) The number of circular permutations of n different objects is (n-1)!

(ii) The number of ways in which n persons can be seated round a table is (n-1)!

(iii) The number of ways in which n persons can be seated round a table is to form a necklace is 12(n-1)!

7. Combinations/Selections:

(i) If n is a natural number and r is a positive integer such that 0≤r≤n, then  nCr=n!(n-r)!r!.

(ii) Selection of r objects out of n distinct objects=Crn.

(iii) Selection of r objects from n distinct objects of which p objects are always included=Cr-pn-p.

(iv) Selection of r objects from n distinct objects of which p objects are never included=Crn-p

(v) Selection of at least one object from n distinct objects=2n-1.

(vi) Selection of at least one object from n objects of which m objects are alike, n objects are alike and rest p are distinct=m+1n+12p-1.

8. Properties related to Crn:

(i) Crn=Cn-rn

(ii) If Cxn=Cyn, then x=y or x+y=n

(iii) Crn+Cr-1n=Crn+1

(iv) CrnCr-1n=n-r+1r

9. Relation between Crn and Prn:

  Prn=Crn·r!

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