Question

The value of –¹⁵C₁ + 2.¹⁵C₂ – 3.¹⁵C₃ + ... –15.¹⁵C₁₅ + ¹⁴C₁ + ¹⁴C₃ + ¹⁴C₅ + ... + ¹⁴C₁₁ is  ______.

पर्याय

• 2¹⁴

• 2¹³ – 13

• 2¹⁶ – 1

• 2¹³ – 14

Answer 1

The value of –¹⁵C₁ + 2.¹⁵C₂ – 3.¹⁵C₃ + ... –15.¹⁵C₁₅ + ¹⁴C₁ + ¹⁴C₃ + ¹⁴C₅ + ... + ¹⁴C₁₁ is `underlinebb(2^13 - 14)`.

Explanation:

Given: S = –¹⁵C₁ + 2.¹⁵C₂ – 3.¹⁵C₃ + ... –15.¹⁵C₁₅ + ¹⁴C₁ + ¹⁴C₃ + ¹⁴C₅ + ... + ¹⁴C₁₁

Let S₁ = –¹⁵C₁ + 2.¹⁵C₂ – ...... –15.¹⁵C_₁₅

= `sum_(r = 1)^15(-1)^r.r.""^15C_r`

= `15sum_(r = 1)^15(-1)^r.""^14C_(r - 1)`

= 15(–¹⁴C₀ + ¹⁴C₁ – ...... –¹⁴C₁₄)

= 15(0) = 0.  ...(∵ –ⁿC₀ + ⁿC₁ – ...... –ⁿC_n = 0)

S₂ = ¹⁴C₁ + ¹⁴C₃ + ... + ¹⁴C₁₁

= (¹⁴C₁ + ¹⁴C₃ + .... + ¹⁴C₁₁ + ¹⁴C₁₃) –¹⁴C₁₃

Since, ⁿC₁ + ⁿC₃ + ⁿC₅ + ........ = 2^n–1

∴ S₂ = 2¹³ – 14

Now, S = S₁ + S₂ = 0 + 2¹³ – 14

∴ S = 2¹³ – 14