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Q.
The number of ten digit numbers that contain only $2$ and $3$ as its digits but no any pairwise $3\, 's$ join together is
Permutations and Combinations
Solution:
When exactly one $3\,'s$ and nine $2\, 's$ are used then numbers possible $={ }^{10} C_{1}$
When exactly two $3'\,s$ and eight $2\,'s$ are used then numbers possible $={ }^{9} C_{2}$
When exactly three $3\, 's$ and seven $2\, 's$ are used then numbers possible $={ }^{8} C_{3}$
When exactly four $3\, 's$ and six $2\,'s$ are used then numbers possible $={ }^{7} C_{4}$
When exactly five $3\, 's$ and five $2\, 's$ are used then numbers possible $={ }^{6} C_{5}$
Adding all $=143$