Find r if (i)⁵P_r : 2⁶P_r-1 (ii) ⁵P_r : ⁶P_r-1

Disclaimer: There should be equal to sign in both as and

(i)

⇒

⇒

⇒

⇒ 42 -13r+r²=12

⇒ r²-13r+30=0

⇒ r²-10r-3r+30=0

⇒ r(r-10)-3(r-10)=0

⇒ (r-3)(r-10)=0

r= 3 or r=10

But r=10 is rejected, as in , r cannot be greater than 5.

Therefore, r=3.

(ii)

⇒

⇒

⇒

⇒ 42 -13r+r²=6

⇒ r²-13r+36=0

⇒ r²-9r-4r+36=0

⇒ r(r-9)-4(r-9)=0

⇒ (r-4)(r-9)=0

r= 4 or r=9

But r=9 is rejected, as in , r cannot be greater than 5.

Therefore, r=4.