Find the value of C r - 1 n C r n .

Given, C r - 1 n C r n = n ! ( r - 1 ) ! [ n - ( r - 1 ) ] ! n ! ( n - r ) ! r ! = 1 ( r - 1 ) ! ( n - r + 1 ) ! × n - r ! r ! = ( n - r ) ! r ( r - 1 ) ! ( r - 1 ) ! ( n - r + 1 ) ( n - r ) ! ∵ n ! = n ( n - 1 ) ! = n ( n - 1 ) ( n - 2 ) ! = ⋯ = r n - r + 1 ∴ C r - 1 n C r n = r n - r + 1

How many terms in the expansion of 3 a + a 3 10 have positive powers of a , how many have negative powers of a ?

Given, 3 a + a 3 10 = 9 + a 2 3 a 10 = 3 a - 10 9 + a 2 10 ( r + 1 ) t h term in the expansion of ( a + b ) n is T r + 1 = C r n a n - r b r ∴ ( r + 1 ) t h term in the expansion of 3 a - 10 9 + a 2 10 is T r + 1 = 3 - 10 a - 10 C r 10 ( 9 ) 10 - r a 2 r = C r 10 ( - 3 ) 10 ( 9 ) 10 - r a 2 r - 10 For positive powers of a , 2 r - 10 > 0 ⇒ 2 r > 10 ⇒ r > 5 C r 10 exists iff 0 ≤ r ≤ 10 ∴ Number of terms having positive powers of a = n { r = 6 , 7 , 8 , 9 , 10 } = 5 ∴ Number of powers having negative powers of a = n { r = 0 , 1 , 2 , 3 , 4 } = 5

Find the middle term(s) in the expansion of a b + b a 6 .

Given, a b + b a 6 Middle term(s) in the expansion of x + y n is n 2 + 1 t h term, if n is even and n + 1 2 t h and n + 3 2 t h terms, if n is odd. ∴ Middle term in the expansion of a b + b a 6 is 6 2 + 1 t h = 4 t h term. r + 1 t h term in the expansion of a + b n is T r + 1 = C r n a n - r b r ∴ Middle term in the expansion of a b + b a 6 = T 4 = C 3 6 a b 6 - 3 b a 3 = 6 × 5 × 4 3 × 2 × 1 a b 3 b a 3 ∵ C r n = n ! r ! ( n - r ) ! = n ( n - 1 ) ( n - 2 ) ⋯ ( n - r + 1 ) r ! ∴ Middle term in the expansion of a b + b a 6 is T 4 = 20 .

Find the middle term(s) in the expansion of x + 1 x 9 .

Given, x + 1 x 9 Middle term(s) in the expansion of x + y n is n 2 + 1 t h term, if n is even and n + 1 2 t h and n + 3 2 t h terms, if n is odd. ∴ Middle terms in the expansion of x + 1 x 9 is 9 + 1 2 t h = 5 t h and 9 + 3 2 t h = 6 t h terms. r + 1 t h term in the expansion of a + b n is T r + 1 = C r n a n - r b r ∴ 5 t h term in the expansion of x + 1 x 9 = T 5 = C 4 9 x 9 - 4 1 x 4 = 9 × 8 × 7 × 6 4 × 3 × 2 × 1 × 5 · 1 x 4 ∵ C r n = n ! r ! ( n - r ) ! = n ( n - 1 ) ( n - 2 ) ⋯ ( n - r + 1 ) r ! = 126 x ∴ 6 t h term in the expansion of x + 1 x 9 is T 6 = C 5 9 x 9 - 5 1 x 5 = 9 × 8 × 7 × 6 × 5 5 × 4 × 3 × 2 × 1 × x 4 · 1 x 5 = 126 x ∴ Middle terms in the expansion of x + 1 x 9 is T 5 = 126 x and T 6 = 126 x .

Find the middle term(s) in the expansion of x 3 2 - y 3 2 8 .

Given, x 3 2 - y 3 2 8 Middle term(s) in the expansion of x + y n is n 2 + 1 t h term, if n is even and n + 1 2 t h and n + 3 2 t h terms, if n is odd. ∴ Middle term in the expansion of x 3 2 - y 3 2 8 is 8 2 + 1 t h = 5 t h term. r + 1 t h term in the expansion of a + b n is T r + 1 = C r n a n - r b r ∴ 5 t h term in the expansion of x 3 2 - y 3 2 8 is T 5 = C 5 8 x 3 2 8 - 5 - y 3 2 5 = 8 ! 5 ! 31 x 3 2 3 - y - 15 2 ∵ C r n = n ! r ! ( n - r ) ! = n ( n - 1 ) ( n - 2 ) ⋯ ( n - r + 1 ) r ! = 8 × 7 × 6 × 5 ! 3 × 2 × 1 × 5 ! × x 9 2 - y - 15 2 ∵ n ! = n ( n - 1 ) ! = n ( n - 1 ) ( n - 2 ) ! = ⋯ = 6 x 9 1 ( - y ) 15 ∴ Middle term in the expansion of x 3 2 - y 3 2 8 is T 5 = 6 x 9 ( - y ) 15 .

Find the 6 t h term in the expansion of x 2 + a 4 y 2 10 .

Given, x 2 + a 4 y 2 10 ( r + 1 ) t ℏ term in the expansion of ( a + b ) n is T r + 1 = C r n a n - r b r ∴ 6 t h term in the expansion of x 2 + a 4 y 2 10 is T 6 = C 5 10 x 2 10 - 5 a 4 y 2 5 = 10 × 9 × 8 × 7 × 6 5 × 4 × 3 × 2 × 1 x 2 a 20 y - 10 ∵ C r n = n ! r ! ( n - r ) ! = n ( n - 1 ) ( n - 2 ) ⋯ ( n - r + 1 ) r ! = 252 x 10 a 20 y - 10 ∴ 6 t h term in the expansion of x 2 + a 4 y 2 10 is 252 x 10 a 20 y - 10 .

Find the fifth term in the expansion of 6 x - a 3 x 10 .

Given, 6 x - a 3 x 10 ( r + 1 ) t ℏ term in the expansion of ( a + b ) n is T r + 1 = C r n a n - r b r ∴ 5 t h term in the expansion of 6 x - a 3 x 10 is T 5 = C 4 10 6 x 10 - 4 - a 3 x 4 = 10 × 9 × 8 × 7 4 × 3 × 2 × 1 ( 6 x ) 6 a 12 x - 4 ∵ C r n = n ! r ! ( n - r ) ! = n ( n - 1 ) ( n - 2 ) ⋯ ( n - r + 1 ) r ! = 210 ( 6 ) 6 x 6 a 12 x ¯ 4 = 210 ( 46656 ) x 2 a 12 = 9797760 x 2 a 12 ∴ 5 t h term in the expansion of 6 x - a 3 x 10 is 9797760 x 2 a 12 .

Is there a term independent of x in 6 x - a 3 x 10 ? If yes find it out.

Given, 6 x - a 3 x 10 ( r + 1 ) t ℏ term in the expansion of ( a + b ) n is T r + 1 = C r n a n - r b r ∴ ( r + 1 ) t ℏ term in the expansion of 6 x - a 3 x 10 is T 6 = C r 10 6 x 10 - r - a 3 x r = C r 10 6 10 - r x 10 - 2 r ( - a ) 3 r If the term is independent of x , then 10 - 2 r = 0 ⇒ r = 5 Yes, there is a term independent of x in the expansion of 6 x - a 3 x 10 and it is ( r + 1 ) t h = ( 5 + 1 ) t h = 6 t h term. ∴ Independent term of x in the expansion of 6 x - a 3 x 10 is T 6 = C 5 10 6 x 5 - a 3 x 5 = 10 × 9 × 8 × 7 × 6 5 × 4 × 3 × 2 × 1 6 5 x 5 - a 15 x - 5 = 252 ( 7776 ) ( - a ) 15 = 1959552 ( - a ) 15