how many 4 letter words can be formed from the word MORADABAD?

MORADABAD has 9 letters out of which 2 are D's, 3 are A's and rest all are distinct.

**(i) Choosing all four distinct letters:**

There are 6 distinct letter is M, O, R, A, D, B, out of which 4 can be chosen in ^{6}C_{4} ways. Each letter can be arranged in 4! ways.

∴ Total number of words = ^{6}C_{4} × 4!

= 360

**(ii) Two distinct and two alike:**

Pairs of alike letters are DD, AAA, out of which one pair can be chosen in ^{2}C_{1} ways.

Two distinct letters out of remaining 5 types of letters can be chosen in ^{5}C_{2} ways.

Total number of ways = ^{2}C_{1} × ^{5}C_{2}

= 240

**(iii) Two alike of one kind and two alike of second kind:**

There are two sets of alike letters, out of these two are to be selected in ^{2}C_{2} ways.

**(iv) Three alike and one distinct:**

There is one set of three alike letters, and it can be selected in one way.

Out of 5 different letters, one can be selected in ^{5}C_{1} ways.

Hence, the total number of 4-letter words = 360 + 240 + 6 + 20

= 626

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