JEE Advanced Level Test: Binomial Theorem- 2 - JEE MCQ

^n+r-1C_r-1

Correct Answer :- a

Explanation : The formula to find the coefficient of a^pb^qc^rd^s in (xa+yb+zc+vd)ⁿ is (n!x^p y^q z^r v^s)/p!q!r!s!

So, coefficient of a⁴b³c²d in (a−b+c−d)¹⁰

= [10!(1)⁴(-1)³(1)²(-1)1]/(4!3!2!1!)

= 12600

Substitute ‘n’ and verify the options.

(1 + a)²ⁿ = C_o + C₁a + C₂a² + … + C_2n a²n
Differentiate both sides w.r.t. ‘a’.

2ⁿ = 4096, find ⁿC_n/2

(n – 2r)² = n + 2

Put x = 1, x  = -1 and then add.

Substitute a positive integer for ‘n’ and verify.

Consider the given function:
c1/c0 + 2c2/c1 +  3c3/c2 + 4c4/c3 +......+ncn/cn−1
= n/1 + [2n(n−1)]/2! . 1/n + [3n(n−1)(n−2)]/3! . 1/[n(n−1)/2!] +......+n.1/n
= n+(n−1)+(n−2)+......1
 = 1+2+3+......+n
 = n(n+1)/2

Substitute ‘n’ and verify the options.

Substitute a positive integer for n and verify.

2ⁿ = 128 ⇒ n = 7

ⁿP_r + r. ⁿP_r-1 = ^{(n + 1)}P_r

n. n! = [(n + 1) – 1]n! = (n + 1)! – n!
∴ 1 + 2! – 1! + … + (n + 1)! – n! = (n + 1)!

Total number of students =10
There are 7 boys and 3 girls.
Seven boys can be arrange in a row  7P7 = 7!ways.
So, 8 places in which we can arrange 3 girls are 8P3 = 8!/(8−3)! = 336ways
The number of arrangement is 7!×336 = k * 8!
= 42

10!/2!

Total number of digits =10
 
i.e. 0,1,2,3,4,5,6,7,8,9
 
require 9 diffrent number 
 
0 cant be placed in first place 
 
= first place can be filled in 9 ways 
 
and the rest 9 blank with 9 digits in 9! ways 
 
total ways = 9×9!

Clearly, one of the odd digits 1, 3, 5, 7, 9 will repeated. The number of selections of the sixth digit = ⁵C₁ = 5.
Required, number of numbers =

Rank is 5(5!) + 0.4! +2(3!) + 1 x 2! + 1 x 1! + 1 = 616

(2 + 3 + 4 + 5)(1111)3! = 93,324

5!. 3!.  2! = 1440

As we know that no. of ways 'n' people can be seated around a round table =(n−1)!
∴ No. of ways '7' people can be seated around a round table =(7−1)!=6!=720
Now, no. of ways if two particular person (out of 7) sit together =5!×2=240
∴ No. of ways 7 people can be seated around a round table if two particular person can not sit together =720−240=480
Hence the correct answer is 480.

⁶P_6-2 = ⁶P₄

In between 6 Pakistanis we have 6 gaps on a circular table, so 7 Indians cannot be arranged in 6 gaps.

ⁿP_r/r