Question

Find the rook polynomial and an expression for the number of matchings of 5 men (rows) with 5 women (columns) given the following forbidden pairings:
(M1,W4), (M2,W2), (M3,W3), (M4,W2), (M4,W4), (M5,W1), (M5,W3), (M5,W45).

Answer is 5! - 8x4! + 21x3! - 20x2!+ 6x1!, please explain how to get it, thanks.

Answer 1

s womeng .whert ere are smens omens Me match, ω1th Wa- Mg matching coithw3 M matching oilh w matching ω'ы, Wa . Mg matching with W matching with w3 M5 matching ω'thW5 Noco we draus a fo tigur that sepresents matching has sorbidodlen Position life タ4. a men who is not malchirg with cwomen thert is no forbidolen

Now we rcarrcnqe the bobi futtin4 together mows wita dakened squave i in the Game colwmn , Puinj togethen for enample1Όώ M2 has dankned gquare inthe曲colum n wz, Similarly Kened quore in oro MA . Kn 6imilaf manner for au dankeneol 2 Nou) to find Ύσ0KPolynomid ot the oON iVen Nou γκ (EB denote the num ber of toa^S to place k capturina rooks on sub board B (ako»ys) no. of darkened square boaxd e

3 thert ase 21 ua^s to choose 2 non captumine Place on boarel B γ3 (B) → Ther are 20 coa45 to cho0H bo ard6 (8 )= 20 Then, art. 6 ωα45 to pla ce on boasd 3 Thore ase no wav to chosge s non- aptaing Place on boaasd Rooko nomial is 1S

Lt 0 -ex 41+2/x31-20x4146 =