Q. 84.5( 6 Votes )

# If ^{8}C_{r} – ^{7}C_{3} = ^{7}C_{2}, find r.

Answer :

Given:

⇒ ^{8}C_{r} – ^{7}C_{3} = ^{7}C_{2}

⇒ ^{8}C_{r} = ^{7}C_{2} + ^{7}C_{3}

We know that ^{n}C_{r} + ^{n}C_{r + 1} = ^{n + 1}C_{r + 1}

⇒ ^{8}C_{r} = ^{7 + 1}C_{2 + 1}

⇒ ^{8}C_{r} = ^{8}C_{3}

We know that if , then one of the following conditions need to be satisfied:

i. p = q

ii. n = p + q

Let us use condition (i),

⇒ r = 3

Let us also check condition (ii),

⇒ 8 = 3 + r

⇒ r = 5

∴ The values of ‘r’ are 3 and 5.

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