A couple decides to have 4 children. If they succeed in having 4 child

THEORY

Permutations of \(n\) things, of which \(P_1\) are alike of one kind, \(P_2\) are alike of a second kind, \(P_3\) are alike of a third kind, ... \(P_r\) are alike of \(r_{th}\) kind such that: \(P_1+P_2+P_3+...+P_r=n\), is:

\(\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}\)

For example, the number of permutations of the letters of the word "gmatclub" is \(8!\), as there are 8 DISTINCT letters in this word.

The number of permutations of the letters of the word "google" is \(\frac{6!}{2!2!}\), as there are 6 letters, with "g" and "o" each appearing twice.

The number of permutations of 9 balls, out of which 4 are red, 3 green, and 2 blue, would be 9!/4!3!2!.

The number of permutations of the letters of the word "ILLUSION" is \(\frac{8!}{2!2!}\), as there are 8 letters, with "L" and "I" each appearing twice.

21. Combinatorics/Counting Methods

• Theory
  Math: Combinatorics
  Combinatorics Made Easy!
  1 hour Combinatorics Video Lesson
  Combinations while dealing with similar and dissimilar things
  When Permutations & Combinations and Data Sufficiency Come Together
  
  • Questions
    Advanced Topic: Probability and Combinations Questions With Solutions
    Letter Arrangements in Words
    Seating Arrangements in a Row and around a Table
    Constructing Numbers, Codes and Passwords
    DS Questions
    PS Questions

For more:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.