যদি ⁿC_r-1 = 36, ⁿC_r = 84  এবং ⁿC_r+1 = 36 হয়,  তবে r এর মান কত?

Explanation:

Another Explanation (5): দেওয়া আছে, ⁿC_r-1 = 36 ...(1) ⁿC_r = 84 ...(2) ⁿC_r+1 = 126 ...(3) এখন, (2) ÷ (1) ⇒ ⁿC_r / ⁿC_r-1 = 84/36 ⇒ [n! / (r! * (n-r)!)] / [n! / ((r-1)! * (n-r+1)!)] = 7/3 ⇒ [(r-1)! * (n-r+1)!] / [r! * (n-r)!] = 7/3 ⇒ (n-r+1) / r = 7/3 ⇒ 3n - 3r + 3 = 7r ⇒ 3n - 10r = -3 ...(4) আবার, (3) ÷ (2) ⇒ ⁿC_r+1 / ⁿC_r = 126/84 ⇒ [n! / ((r+1)! * (n-r-1)!)] / [n! / (r! * (n-r)!)] = 3/2 ⇒ [r! * (n-r)!] / [(r+1)! * (n-r-1)!] = 3/2 ⇒ (n-r) / (r+1) = 3/2 ⇒ 2n - 2r = 3r + 3 ⇒ 2n - 5r = 3 ...(5) এখন, (4) * 2 - (5) * 3 ⇒ (6n - 20r) - (6n - 15r) = -6 - 9 ⇒ -5r = -15 ⇒ r = 3 সুতরাং, r এর মান 3। ফাইনাল আনসার : r = 3 🥳