2022_IAS_Qn A77. There are 9 cups placed on a table arranged in equal number of rows and columns

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There are 9 cups placed on a table arranged in equal number of rows and columns out of which 6 cups contain coffee and 3 cups contain tea. In how many ways can they be arranged so that each row should contain at least one cup of coffee?
(a) 18
(b) 27
(c) 54
(d) 81

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At least one cup of coffee ☕ required in each row.
Possible arrangements are.
(3, 2, 1) and (2, 2, 2)
Case 1a, we are arranging three coffees in 1st row, 2 coffees in second raw and 1 coffee in third raw.
Ie.,
☕ ☕☕
☕☕ 🍵
☕ 🍵 🍵
Case 1b
☕ ☕☕
☕🍵 🍵
☕ ☕ 🍵
Then, three cups of coffee in a raw can be arranged in only one way.
Two cups of coffee ☕ in a row can be arranged in three ways.
Ie.,
☕ ☕ 🍵
☕🍵 ☕
🍵 ☕ ☕
One cup of coffee in a raw can be arranged in three ways.
Ie.,
☕ 🍵 🍵
🍵 🍵 ☕
🍵☕ 🍵

Or
(3, 2, 1) can be arranged in 3!=6 way
In each case 1 * 3 *3=9 arrangements possible.
Total arrangements = 6*9=54.
And (2, 2, 2)
Two cups of coffee ☕ in a row can be arranged in three ways.
Total arrangements = 3*3*3=27
Grand total arrangements =54+27=81

AbdusCSATMATH

Sir shouldn't we have taken permutation instead of combination because say 1 cup of coffee in a row can be placed in any one of the three slots, like CTT, TTC, TCT.. ...so we have the options to place it in any of the one slot...so order also matters here isn't it ?

poorviasthana

Sir i didn't not understand
case b

pavanashreeng

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