If 15C3r = 15Cr + 3, find r.

1.	If 15C3r = 15Cr + 3, find r.
Answer» Given, ¹⁵C_3r = ¹⁵C_r+3 We know that, If ⁿC_P = ⁿC_q, Then one of the following conditions need to be satisfied : i. p = q ii. n = p + q Let us use condition (i), ⇒ 3r = r + 3 ⇒ 2r = 3 ⇒ r = \(\frac{3}{2}\) ⇒ r + 3 = \(\frac{3}{2}\) + 3 ⇒ r + 3 = \(\frac{9}{2}\) We know that, For a combination ⁿC_r,r≥0 and r should be an integer, which is not satisfies here, So, the condition (ii) must be satisfied, ⇒ 15 = 3r + r + 3 ⇒ 4r = 12 ⇒ r = \(\frac{12}{4}\) ⇒ r = 4 ∴ The value of r is 4.

Answer»

Given,

¹⁵C_3r = ¹⁵C_r+3

We know that,

If ⁿC_P = ⁿC_q,

Then one of the following conditions need to be satisfied :

i. p = q

ii. n = p + q

Let us use condition (i),

⇒ 3r = r + 3

⇒ 2r = 3

⇒ r = \(\frac{3}{2}\)

⇒ r + 3 = \(\frac{3}{2}\) + 3

⇒ r + 3 = \(\frac{9}{2}\)

We know that,

For a combination ⁿC_r,r≥0 and r should be an integer, which is not satisfies here,

So, the condition (ii) must be satisfied,

⇒ 15 = 3r + r + 3

⇒ 4r = 12

⇒ r = \(\frac{12}{4}\)

⇒ r = 4

∴ The value of r is 4.

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