Statement I n C0+ n C2+ n C4+ ldots ldots=2n-1 Statement II n C1+ n C3+ n C5+ ldots ldots=2n-1

Question

Statement I n C0+ n C2+ n C4+ ldots ldots=2n-1 Statement II n C1+ n C3+ n C5+ ldots ldots=2n-1 (A) Statement I is true; Statement II is true; Statement II is a correct explanation for Statement I (B) Statement I is true; Statement II is true; Statement II is not a correct explanation for Statement I (C) Statement I is true; Statement II is false (D) Statement I is false; Statement II is true

Tardigrade Admin · Accepted Answer

In the binomial expression, we have (a+b)n= n C0 an+ n C1 an-1 b+ n C2 an-2 b2+ ldots+ n Cn bn .....(i) The coefficients n C0, n C1, n C2, ldots ., n Cn are known as binomial or combinatorial coefficients. Putting a=b=1 in (i), we get n C0+ n C1+ n C2+ ldots+ n Cn=2n Thus, the sum of all binomial coefficients is equal to 2n. Again , putting a=1 and b=-1 in Eq. (i), we get n C0+ n C2+ n C4+ ldots= n C1+ n C3+ n C5+ ldots Thus, the sum of all the odd binomial coefficients is equal to the sum of all the even binomial coefficients and each is equal to (2n/2)=2n-1. ⇒ n C0+ n C2+ n C4+ ldots= n C1+ n C3+ n C5+ ldots=2n-1

Q. Statement I nC0​+nC2​+nC4​+……=2n−1 Statement II nC1​+nC3​+nC5​+……=2n−1

Statement I is true; Statement II is true; Statement II is a correct explanation for Statement I

Statement I is true; Statement II is true; Statement II is not a correct explanation for Statement I

Statement I is true; Statement II is false

Statement I is false; Statement II is true

Q. Statement I $^{n} C_{0} +^{n} C_{2} +^{n} C_{4} + \dots\dots = 2^{n - 1}$
Statement II $^{n} C_{1} +^{n} C_{3} +^{n} C_{5} + \dots\dots = 2^{n - 1}$