Home
Class 12
MATHS
Using binomial theorem (without using th...

Using binomial theorem (without using the formula for `.^n C_r`) , prove that `.^nC_4+^m C_2-^m C_1.^n C_2 = .^m C_4-^(m+n)C_1.^m C_3+^(m+n)C_2.^m C_2-^(m+n)C_3^m.C_1 +^(m+n)C_4dot`

Text Solution

Verified by Experts

`.^(m)C_(4)-.^(m+n)C_(1).^(m)C_(3)+.^(m+n)C_(2).^(m)C_(2)-.^(m+n)C_(3).^(m)C_(1)+.^(m+n)C_(4)`
`= .^(m+n)C_(0).^(m)C_(4)-.^(m+n)C_(1).^(m)C_(3)+.^(m+n)C_(2).^(m)C_(2)-.^(m+n)C_(3).^(m)C_(1)+.^(m+n)C_(4).^(m)C_(0)`
= Coefficient of `x^(4)` in `(1+x)^(m+n)(1-x)^(m)`
= Coefficient of `x^(4)` in `(1-x)^(m)(1+x)^(n)`
= Coefficient of `x^(4)` in `[1-.^(m)C_(1)x^(2)+.^(m)C_(2)x^(4)-"....."][1+.^(n)C_(1)x+.^(n)C_(2)x^(2)+"....."+.^(n)C_(n)x^(n)]`
`= .^(n)C_(4)-.^(m)C_(1) xx .^(n)C_(2)+.^(m)C_(2)`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Example|10 Videos

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Concept Application Exercise 8.1|17 Videos

  • AREA

    CENGAGE ENGLISH|Exercise Comprehension Type|2 Videos

  • CIRCLE

    CENGAGE ENGLISH|Exercise MATRIX MATCH TYPE|7 Videos

Similar Questions

Explore conceptually related problems

Prove that ^m C_1^n C_m-^m C_2^(2n)C_m+^m C_3^(3n)C_m-=(-1)^(m-1)n^mdot

Show that: ^mC_1+^(m+1)C_2+^(m+2)C_3+…….+^(m+n-1)C_n=^nC_1+^(n+1)C_2+^(n+2)C_3+…+^(n+m-1)C_m .

If m in N and mgeq2 prove that: |1 1 1\ ^m C_1\ ^(m+1)C_1\ ^(m+2)C_1\ ^m C_2\ ^(m+1)C_2\ ^(m+2)C_2|=1 .

Prove that .^(n)C_(0) +5 xx .^(n)C_(1) + 9 xx .^(n)C_(2) + "…." + (4n+1) xx .^(n)C_(n) = (2m+1) 2^(n) .

[(^nC_0+^n C_3+)-1/2(^n C_1+^n C_2+^n C_4+^n C_5]^2 + 3/4(^n C_1-^n C_2+^n C_4 +)^2= a. 3 b. 4 c. 2 d. 1

Prove that 1/(n+1)=(.^n C_1)/2-(2(.^n C_2))/3+(3(.^n C_3))/4- . . . +(-1^(n+1))(n*(.^n C_n))/(n+1) .

If alpha=^m C_2,t h e n^(alpha)C_2 is equal to a. ^m+1C_4 b. ^m-1C_4 c. 3^(m+2)C_4 d. 3^(m+1)C_4

((1 + ""^nC_1 + ""^nC_2 + ""^nC_3+…….+nC_n)^2)/(1 + ""^(2n)C_1 + ""^(2n)C_2 + ""^(2n)C_3 + ……… + ""^(2n)C_(2n)) =

Statement 1: a fair coin tossed 15 times, then the probability of getting head as many times in the first ten throws as in the last five is 3003/32768. Statement 2: Sum of the series ^m C_r^n C_0+^m C_(r-1)^n C_1++^m C_0^n C_r=^(m+n)C_r .

If (1+x)^n=sum_(r=0)^n C_r x^r , then prove that C_1+2C_2+3C_3+....+n C_n=n2^(n-1)dot .