Question

2.Determine n if() 2C3 C3 12: 1) 2C3 C11:1

Answer 1

(i)

The given equation is C 2n 3 : C n 3 =12:1.

The ratio means the fraction of the terms, the given equation is written as,

C 2n 3 C n 3 = 12 1

The formula to calculate the combination is,

C n r = n! ( n−r )!r!

Where,n is the number of objects taken r at a time.

Use the formula of combination to find the value of n.

( 2n )! ( 2n−3 )!3! n! ( n−3 )!3! = 12 1

Cancel the common factors and cancel the common factors by factorizing the bigger term to the factorial.

The formula to calculate the factors of a factorial in terms of factorial itself is,

n!=n( n−1 )! n!=n( n−1 )( n−2 )!                 [ n≥2 ] ⋮

The above equation becomes,

( 2n )( 2n−1 )( 2n−2 )( 2n−3 )! ( 2n−3 )! n( n−1 )( n−2 )( n−3 )! ( n−3 )! = 12 1 2( 2n−1 )( 2n−2 ) ( n−1 )( n−2 ) = 12 1 2( 2 )( 2n−1 )( n−1 ) ( n−1 )( n−2 ) = 12 1 ( 2n−1 ) ( n−2 ) = 3 1

Cross multiply both sides,

2n−1=3( n−2 ) 2n−1=3n−6 n=5

Thus, the value of n is 5.

(ii)

The given equation is C 2n 3 : C n 3 =11:1.

The ratio means the fraction of the terms, thus the given equation is written as,

C 2n 3 C n 3 = 11 1

The formula to calculate the combination is,

C n r = n! ( n−r )!r!

Where,n is the number of objects taken r at a time.

Use the formula of combination to find the value of n.

( 2n )! ( 2n−3 )!3! n! ( n−3 )!3! = 11 1

Cancel the common factors and cancel the common factors by factorizing the bigger term to the factorial.

The formula to calculate the factors of a factorial in terms of factorial itself is,

n!=n( n−1 )! n!=n( n−1 )( n−2 )!                 [ n≥2 ] ⋮

The above equation becomes,

( 2n )( 2n−1 )( 2n−2 )( 2n−3 )! ( 2n−3 )! n( n−1 )( n−2 )( n−3 )! ( n−3 )! = 11 1 2( 2n−1 )( 2n−2 ) ( n−1 )( n−2 ) = 11 1 2( 2 )( 2n−1 )( n−1 ) ( n−1 )( n−2 ) = 11 1 4( 2n−1 ) ( n−2 ) =11

Cross multiply both sides,

4( 2n−1 )=11( n−2 ) 8n−4=11n−22 22−4=11n−8n 3n=18

Further simplify the above equation,

n= 18 3 n=6

Thus, the value of n is 6.