Q. Prove the following 
  1. P(n,n) = 2P (n,n-2)
  2. P(n,r) = P(n-1, r) + r , P(n-1, r-1)
  3. ​P(n, n) = P(n, n-1)
  4. P(n,r) = n.P(n-1,r-1)

Dear student
iv)
n Pn-1,r-1=nn-1!n-1-r-1!=n!n-r!=Pn,ri) 2P n,n-2=2n!n-n-2!=2n!2!=n!=Pn,n
ii)Pn,n=Pn,n-1Pn,n-1=n!n-n-1!=n!1!=n!=Pn,niii) Pn,r=Pn-1,r+r Pn-1,r-1Pn-1,r+r Pn-1,r-1=n-1!n-1-r!+r n-1!n-1!-r-1!=n-1!n-r-1!+r n-1!n-r!=n-1!n-r-1!+r n-1!n-rn-r-1!=n-1!n-r-1!1+rn-r=n-1!n-r-1!n-r+rn-r=n-1!n-r-1!nn-r=n!n-r!=Pn,r
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