Question

The number of ten digit numbers that contain only $2$ and $3$ as its digits but no any pairwise $3{}^{\prime }s$ join together is

• A. 145
• B. 143
• C. 129
• D. None

Answer 1

Answer: (B)

When exactly one $3{}^{\prime }s$ and nine $2{}^{\prime }s$ are used then numbers possible $={}^{10}{C}_1$
When exactly two $3^{\prime }s$ and eight $2{}^{\prime }s$ are used then numbers possible $={}^9{C}_2$
When exactly three $3{}^{\prime }s$ and seven $2{}^{\prime }s$ are used then numbers possible $={}^8{C}_3$
When exactly four $3{}^{\prime }s$ and six $2{}^{\prime }s$ are used then numbers possible $={}^7{C}_4$
When exactly five $3{}^{\prime }s$ and five $2{}^{\prime }s$ are used then numbers possible $={}^6{C}_5$
Adding all $=143$