Questions - Answers

 

2. Find the number of ways of arranging 6 boys and 5 girls in a row so that

    (i) all the girls sit together                      

   (ii) no two girls sit together

   (iii) boys and girls sit alternately and     

   (iv) no two boys sit together.

Ans: (i)  

     6 boys + 5 girls (1 unit) = 7 to be adjusted in ways.

     But 5 girls among themselves can be adjusted in ways.                             

 total no. of ways =    .  

(ii)  

    no. of ways boys are adjusted =

    no. of ways girls are adjusted =7P_⁵             

      total no. of ways = . 7P₅

(iii)  

     no. of ways boys and girls sit alternately =  

 . 

C    X    X    X    X    X    X         →   = 720

I    X     X    X    X    X    X         →   = 720

O   X     X    X    X    X    X         →   = 720

R   X     X    X    X    X    X         →   = 720

T   X     X    X    X    X    X         →   = 720

VC    X    X    X    X    X             →  = 120

VICO   X    X    X                       →    = 006

VICR   X    X    X                       →    = 006

VICTORY                                  →    = 001

                                                 Total :  3733     

∴ Rank of the word 'VICTORY' is 3733.

5. Find the number of ways of arranging the letters of the word 'INDEPENDENCE'.

Ans: INDEPENDENCE contains in all 12 letters of which I occurs 1 time, N occurs 3 times, D occurs 2 times, E occurs 4 times, P occurs 1 time and C occurs 1 time.

6. If nP_r = 5040 and nC_r = 210. Find 'r' and 'n'.

Ans: nP_r =  . nC_r    5040 =  . 210

       24 =   

      ∴ r = 4

        nP₄ = 5040

               = 10.9.8.7

          ∴ n = 10

7. Prove that for 3 ≤ r ≤ n,

    (n-3)C_r + 3. (n-3) C_(r-1) + 3. (n-3) C_(r-2) + (n-3) C_(r-3) = nC_r

Ans:

L.H.S = (n-3)C_r + (n-3) C_r−1 + 2. [(n-3) C_(r-1) + (n-3) C_(r-2) ]+ (n-3) C_(r-2) + (n-3) C_(r-3)

          = (n-2) C_r + 2 [(n-2) C_(r-1)] + (n-2) C _(r-2)

          = [(n-2) C_r + (n-2) C_(r-1)] + [(n-2) C_(r-1) + (n-2) C_(r-2) ]

          = (n-1) C_r + (n-1) C _(r-1)

          = nC_r  = R.H.S.

8.Find the number of ways of forming a committee of 5 members out of 6 Indians and 5 Americans so that the Indians are always in the majority in the committee.

 ∴  Total no. of ways = 6C₃ . 5C₂ + 6C₄ .5C₁+ 6C₅ . 5C₀

= (20) (10) + (15) (5) + (6) (1) =  200 + 75 + 6  =  281
 

9. Prove: 4n C_2n : 2n C_n = [1.3.5....(4n-1)] :  [1.3.5... (2n-1)]²

         

        ∴ 4n C_2n : 2n C_n =  [1.3.5.... (4n-1)]: [1.3.5....(2n-1)]²
 

10. Find the number of diagonals of a polygon of 12 sides.
 

11. Find the number of positive divisors of 10800 other than 1 and the number itself.

Ans: 10800 = 5². 3³. 2⁴

No. of divisors required = (2+1) (3+1) (4+1) - 2  = 60 - 2 = 58.