HOW TO FIND THE VALUE OF N AND R IN COMBINATION

How to Find the Value of n and r in Combination :

In this section, we will learn, how to find the values of n and r in the given combination. 

How to Find the Value of n and r in Combination - Questions

Question 1 :

If nC12 = nC9 find 21Cn.

Solution :

nC12  =  n!/(n - 12)! 12!  -----(1)

nC9  =  n!/(n - 9)! 9!     -----(2)

(1)  =  (2)  

n!/(n - 12)! 12!  =  n!/(n - 9)! 9!

(n - 9)! 9!  =  (n - 12)! 12!

(n - 9)(n - 10)(n - 11)(n - 12)! 9!  =  (n - 12)! 12 ⋅ 11 ⋅ 10 ⋅ 9!

(n - 9)(n - 10)(n - 11)  =  12 ⋅ 11 ⋅ 10 

n - 9  =  12

n  =  12 + 9

n  =  21

 21Cn =   21C21  =  1

Hence the answer is 1.

Question 2 :

If 15C2r−1 = 15C2r+4, find r.

Solution :

If nC=  nCy  ==>  x  =  y (or) x + y  =  n

2r - 1 + 2r + 4  =  15

4r + 3  =  15

4r  =  12

Divide by 4 on both sides.

r  =  12/4  ==>  3

Hence the value of r is 3.

Question 3 :

If nPr = 720, and nCr = 120, find n, r.

Solution :

nPr = 720

n!/(n - r)!  =  720  ---(1)

nCr = 120

n!/(n - r)! r!  =  120  ---(2)

Divide (1) by (2), we get 

r!  =  720/120

r!  =  6

r!  =  3!  ==> r = 3

By applying the value of r in the (1), we get

n!/(n - 3)!  =  720 

n(n - 1) (n - 2)  =  720

n(n - 1) (n - 2)  =  10 ⋅ 9 ⋅ 8

n  =  10

Hence the value of r and n are 3 and 10 respectively.

Question 4 :

Prove that 15C3 + 2 × 15C4 + 15C5 = 17C5.

Solution :

L.H.S 

  =   15C3 + 2 × 15C4 + 15C5

  =   15C3 + 15C4 + 15C15C5

=   15C4 15C3 +  15C15C4

By using the property :

nCr + n Cr−1 = n+1Cr

=  16C4 16C5

=  17C5 ---> R.H.S

Question 5 :

Prove that 35C5 + ∑ 4r=0 (39−r)C4  =  40C5.

Solution :

L.H.S

 =  35C5 + ∑ 4r=0 (39−r)C4

  =  35C5 + 39C438C37C36C35C4

By using the property :

nCr + n Cr−1 = n+1Cr

  =  35C5 +  35C4 + 39C438C37C36C

  =  36C36C4 39C438C37C

  =  37C37C39C438C

=  38C+ 38C 39C

=  39C 39C

=  40C

After having gone through the stuff given above, we hope that the students would have understood, how to find the values of n and r in the given combinations

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. How to Solve Age Problems with Ratio

    Mar 28, 24 02:01 AM

    How to Solve Age Problems with Ratio

    Read More

  2. AP Calculus BC Integration of Rational Functions by Partical Fractions

    Mar 26, 24 11:25 PM

    AP Calculus BC Integration of Rational Functions by Partical Fractions (Part - 1)

    Read More

  3. SAT Math Practice

    Mar 26, 24 08:53 PM

    satmathquestions1.png
    SAT Math Practice - Different Topics - Concept - Formulas - Example problems with step by step explanation

    Read More